Weak Pullbacks of Topological Groupoids

WEAK PULLBA CKS OF TOPOLOGICAL GROUPOIDS A. CENSOR AND D. GRANDINI Abstract. W e int ro duce the catego ry HG , who se o b jects are top ological group oids en- dow ed with compatible measur e theor etic data: a Haa r sys tem a nd a measur e on the unit space. W e then define and stud y the notion o f w eak-pullback in the category of to polo g- ical group oids, and subsequently in HG . The category H G is the setting for topo logical group oidification, which w e presen t in separate pap ers, and in which the w eak pullback is a key ingredient. 1. Introduction The leading actors in this paper are g roup oids tha t w e call H aar gr oup oids 1 . A Haa r group oid is a top ological group oid endo wed with certain compat ible measure theoretic in- gredien ts. More precisely , a Haar group o id is a lo cally compact, second coun table, Hausdorff group oid G , whic h admits a contin uous left Haar system λ • , and is equipp ed with a non-zero Radon measure µ (0) on its unit space G (0) , suc h that µ (0) is quasi-in v ariant with resp ect to λ • . Maps betw een Haar group oids are con tin uous group oid homomorphisms, whic h resp ect the extra structure in an appropriate sense. One is nat ur a lly led to de fine a category , whic h w e denote b y HG , the category of Haar group oids. Section 2 in tro duces this cat ego ry . A general study of the category H G f r om a purely categorical p ersp ectiv e will b e presen ted in a separate pa p er. In this pa p er w e fo cus on one sp ecific cat ego rical notion, namely the we ak pul lb ack . W e first construct the w eak pullbac k of top o logical group oids. The w eak pullbac k of the f ollo wing giv en cospan diagra m of top olog ical group oids a nd con tinuous homomorphisms: S p   ? ? ? ? ? ? ? T q   ~ ~ ~ ~ ~ ~ ~ G is a t op ological gro up o id P along with pro jections π S : P → S and π T : P → T , whic h together giv e rise to the follo wing diagram (which do es n ot comm ute): P π S          π T   @ @ @ @ @ @ @ S p   ? ? ? ? ? ? ? T q   ~ ~ ~ ~ ~ ~ ~ G Date : Novem b er 8, 2 018. 2010 Mathematics Subje ct Classific ation. 22A22 ; 28A50. Key wor ds and phr ases. Group oid; Haar group oid; W eak pullback; Haar sys tem; quasi inv arian t measure; disintegration. 1 A discussion r egarding ter minology app ears at the end of this introduction. 1 2 A. CENS OR A ND D. GRA NDINI As a set, P is con tained in the cartesian pro duct S × G × T , from wh ic h it inherits its top ology . The elemen ts of P are triples of the fo r m ( s, g , t ), w here p ( s ) and q ( t ) are not equal to g , but rather in the same orbit of G via g . More precisely , denoting the range and source maps o f G b y r G and d G resp ectiv ely , P := { ( s, g , t ) | s ∈ S, g ∈ G, t ∈ T , r G ( g ) = r G ( p ( s )) a nd d G ( g ) = r G ( q ( t )) } . The group oid structure of P is describ ed in Section 3, follow ed by a discussion o f its prop- erties. In the discrete group o id setting, our notion of w eak pullbac k reduces to the one in tro duced b y Baez et al. in [2 ], whic h in turn generalizes the more familiar notion of a pullbac k in the category of sets. Upgrading the w eak pullbac k f rom top ological group oids to the category H G requires non- trivial measure theory and ana lysis. In Se ction 4 w e construct a Haar system for P . Section 5 is then devoted to creating a quasi in v arian t measure on P (0) . Finally , in Section 6, w e pro ve that with these additional ingredie n ts, sub ject to a certain additional a ssumption, w e indeed obtain a w eak pullbac k in HG . This pap er is part of a pro ject w e are currently w orking on, in whic h w e are extending group oidification from the discrete setting to the realm of top olog y and measure theory . Gr oup oidific ation is a form o f categorification, in tro duced b y John Baez and James Dolan. It has b een successfully applied to sev eral structures , whic h include F e ynman D iagrams, Hec ke Algebras and Hall Algebras. An excellen t accoun t of group oidification a nd its triumphs to date can b e found in [2]. So f ar, the scop e of group oidification and its in v erse pro cess of degroup oidification has b een limited to purely algebraic structures and discrete g roup oids. The category HG provides the setting for our attempt at top olog ical gro upo idification, in whic h the notion of the w eak pullbac k pla ys a vital ro le. This line of researc h is pursued in separate papers. This pap er relies hea vily on general top ological and me asure theoretic tec hniques related to Borel and con tin uous systems o f measures and t heir mapping pro p erties. A detailed study of this necessary bac kground theory app ears in our paper [4], from which w e quote man y definitions and res ults and to whic h we make fr equen t references throughout this text. 1.1. A note ab out terminology. Seeking a distinctiv e name for the group oids w e consider in these n otes a nd in our subs equen t w ork on t o p o logical group oidification, w e o pted to call them “Haar group oids”. These gro upo ids b ear close res em blance to me asur e gr oup oids with Haar me asures, as studied b y P eter Hahn in [5], following Mack ey [6] and Ramsay [10], leading to the theory o f gro up o id von-Neumann algebras. Like the group oids w e consider, measure group oids carry a measure (or measure class), whic h a dmits a disinte gration via the range map, namely what is now ada ys kno wn as a Haar s ystem. The main discrepancies are that w e require our group oids to exhibit a nice to p o logy (lo cally compact, Hausdorff ) and to b e endow ed with a c ontinuous Haa r system, whereas measure group oids need only hav e a Borel structure in general, and host Borel Haar systems. Lo cally compact topo logical group oids whic h ma y admit con tinuous Haar sy stems are as w ell studied in the literature a s measure g r o up o ids, in par ticular as part of group oid C ∗ - algebra theory as dev elop ed b y Jean Renault in [11] (other standard references include [7] and [8]). In man y cases lo cally compact group oids indeed exhibit the full structure of our Haar group o ids, yet the literature do es not single them out terminology- wise. WEAK PULLBAC KS OF TOPOLOGICAL GR OUPOI DS 3 2. Preliminaries and the ca tegor y H G W e begin b y fixing notatio n. W e shall denote the unit sp ac e of a group oid G b y G (0) and the set of c omp osable p airs b y G (2) . The r ange (or target) and domain (or source) maps of G are denoted resp ectiv ely b y r a nd d , or b y r G and d G when disam biguation is necessary . W e set G u = { x ∈ G | r ( x ) = u } , G v = { x ∈ G | d ( x ) = v } and G u v = G u ∩ G v , for all u, v ∈ G (0) . Th us G u u is the isotr opy gr oup at u . W e le t G = G (0) /G = { [ u ] | u ∈ G (0) } denote the orb i t sp ac e of a group oid G . The orbit space G inherits a top ology from G via G (0) , defined by declaring W ⊆ G to be op en whenev er q − 1 ( W ) is op en in G (0) , where q : G (0) − → G is the quotien t map u 7→ [ u ]. Thr oughout this p ap er, w e wi l l assume our top olo gic al gr oup oids to b e s e c ond c ountable, lo c al l y c omp act and Hausdorff. Any suc h group oid G is metrizable and normal, and satis- fies that ev ery lo cally finite measure is σ -finite. Moreo ve r, G is a P o lish space and hence strongly Radon, i.e. ev ery lo cally finite Borel measure is a Radon measure. F or more on P olish groupoids, w e refer the reader to a p ap er b y Ramsa y [9]. In g eneral, ho wev er, G do es not necess arily inherit these properties, a fact that will require o ccasional extra caution. Haar syste ms for group oids pla y a k ey role in this pap er. In the group oid literature, mo dulo minor discre pancies b et w een v arious sources (see for example standard reference s suc h as [7 ], [8], [11] and [1]), a c on tinuous left Haar s ystem is usually defined to b e a family λ = { λ u : u ∈ G (0) } of p ositiv e (Ra don) measures on G satisfy ing the follo wing prop erties: (1) supp ( λ u ) = G u for ev ery u ∈ G (0) ; (2) for an y f ∈ C c ( G ), the function u 7→ R f dλ u on G (0) is in C c ( G (0) ); (3) for an y x ∈ G and f ∈ C c ( G ), R f ( xy ) dλ d ( x ) ( y ) = R f ( y ) d λ r ( x ) ( y ) . In this pap er we shall use D efinition 2.1 b elow as our definition of a Haar system. It is tak en from [4], where it is sho wn to b e equiv alen t to the more common definition ab ov e. F or the con ve nience of the reader w e include here a v ery brief summary of the notions from [4] that lead to Definition 2.1, a ll of which we will use extensiv ely throughout t his pap er. Henc eforth, as in [4], a ll top ological spaces a re assume d t o b e second countable and T 1 in general, a nd also lo cally c ompact and Hausdorff whenev er dealing with con tinuous systems of measures . Let π : X → Y b e a Borel map. A s ystem of me asur es ([4], Definition 2.2) on π is a family of (p ositiv e, Borel) measures λ • = { λ y } y ∈ Y suc h that: (1) Eac h λ y is a Borel measure on X ; (2) F or ev ery y , λ y is concen trated on π − 1 ( y ). W e will denote a map π : X → Y admitting a s ystem of measures λ • b y the diagra m X π λ • / / Y . W e will say that a system of measures λ • is p ositive on op en sets ([4], Definition 2.3) if λ y ( A ) > 0 for ev ery y ∈ Y and fo r ev ery op en set A ⊆ X suc h that A ∩ π − 1 ( y ) 6 = ∅ . A system of measures λ • on a contin uous map π : X → Y w ill b e called a c ontinuous system of me asur es or CS M ([4], Definition 2.5) if f or ev ery non-negative con tinuous compactly supp orted function 0 ≤ f ∈ C c ( X ), the map y 7→ R X f ( x ) dλ y ( x ) is a con tin uous function on Y . A system of measures λ • on a Borel map π : X → Y is called a B or el system of m e asur es or BSM ([4], Definition 2.6 ) if for ev ery Borel subset E ⊆ X , the function λ • ( E ) : Y → [0 , ∞ ] giv en b y y 7→ λ y ( E ) is a Bor el function. A system of measures λ • satisfying that ev ery x ∈ X 4 A. CENS OR A ND D. GRA NDINI has a neighborho o d U x suc h that λ y ( U x ) < ∞ for ev ery y ∈ Y , will b e called lo c al ly fi n ite ([4], D efinition 2.14), and lo c al ly b ounde d if there is a constan t C x > 0 suc h that λ y ( U x ) < C x for an y y ∈ Y ([4], Definition 2.3). A detailed discussion of the mutual relations b et w een the ab ov e concepts a pp ear s in [4]. Let G b e a to p o logical group oid. A system o f measures λ • on the range map r : G → G (0) is said to b e a system of me asur es on G ([4], Definition 7.1) . It is called left invariant ([4], Definition 7.2) if for ev ery x ∈ G and for ev ery Borel subset E ⊆ G , λ d ( x ) ( E ) = λ r ( x )  x · ( E ∩ G d ( x ) )  . Definition 2.1. ( [4] , De finition 7 . 5 ) A c ontinuous le f t Haar system for G is a system of me asur es λ • on G which is c ontinuous, left invaria nt and p ositive o n op en sets. Pla ying side by side to the Haar system λ • , another leading actor in our w or k is a Radon measure on the unit space G (0) of a group oid G , which w e denote by µ (0) . The measure µ (0) will b e relat ed to λ • via the notion of quasi inv aria nce, whic h w e sp ell out b elo w. W e usually follo w [7], where the reader can find m uch more ab out t he imp ortant role of quasi in v arian t measures in gro up o id theory . Definition 2.2. L et G b e a gr oup oid a d mitting a Haar system λ • and a R ado n me asur e µ (0) on G (0) . T h e induc e d me asur e µ o n G is define d for any Bor el set E ⊆ G by the formula: µ ( E ) = Z G (0) λ u ( E ) dµ (0) ( u ) . Lemma 2.3. The ind uc e d me asur e µ is a R adon me asur e on G . Pr o of. Since G is strongly Radon, it suffices to pro v e that µ is lo cally finite. The induced measure µ is obta ined as a comp osition of the system λ • with the measure µ (0) . The Haar system λ • is a CSM, hence a lo cally b ounded BSM, by Lemma 2.11 and Prop osition 2.2 3 of [4]. In a dditio n, the measure µ (0) is lo cally finite. Therefore, the conditions of Corollary 3.7 in [4] are met, and w e conclude that µ is lo cally finite.  The follow ing simple observ ation will b e useful in t he sequel. Lemma 2.4. F or any Bor el function f on G : Z G f ( x ) dµ ( x ) = Z G (0)  Z G f ( x ) dλ u ( x )  dµ (0) ( u ) . Pr o of. F or ev ery Borel subset E ⊆ G , by Definition 2.2, Z G χ E ( x ) dµ ( x ) = µ ( E ) = Z G (0) λ u ( E ) dµ (0) ( u ) = Z G (0)  Z G χ E ( x ) dλ u ( x )  dµ (0) ( u ) . Generalizing from χ E to an y Borel function f is routine.  The image of µ under inv ersion is defined by µ − 1 ( E ) := µ ( E − 1 ) = µ ( { x − 1 | x ∈ E } ) for an y Borel set E ⊆ G . Remark 2.5. It is a standa r d exer cise to show that fo r any B or el function f , Z G f ( x ) dµ − 1 ( x ) = Z G f ( x ) dµ ( x − 1 ) . WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 5 Definition 2.6. L et G b e a gr oup oid admitting a Haar syst em λ • and a R adon me asur e µ (0) on G (0) . T h e me asur e µ (0) is c al le d quasi invariant if the induc e d me asur e µ satisfie s µ ∼ µ − 1 . Here ∼ denotes equiv alence of measures in t he sense of b eing m utually absolutely con tinuous. Remark 2.7. L et µ (0) b e quasi i n variant. The R adon-Niko dym derivative ∆ = dµ/dµ − 1 is c al le d the mo dular function o f µ . Although ∆ is d e termine d onl y a.e., it c an b e c h o sen ( [7 ] , The or em 3.15) to b e a homomorphis m fr om G to R × + , so we w i l l assume this to b e the c ase. R e c al l that for any Bor el function f , (1) Z G f ( x ) dµ ( x ) = Z G f ( x )∆( x ) dµ − 1 ( x ) . F u rthermor e, ∆ − 1 = dµ − 1 /dµ satisfies the useful formula (2) Z G f ( x )∆ − 1 ( x ) dµ ( x ) = Z G f ( x − 1 ) dµ ( x ) , sinc e R G f ( x )∆ − 1 ( x ) dµ ( x ) = R G f ( x ) dµ − 1 ( x ) = R G f ( x ) dµ ( x − 1 ) = R G f ( x − 1 ) dµ ( x ) by R e- mark 2.5. Definition 2.8. L et G b e a top olo gic al gr oup oid, wh i c h satisfies the fol lowing as s umptions: (1) The top olo gy of G is l o c al ly c omp act, se c ond c ountable and Hausdorff. (2) G adm i ts a c ontinuous left Haar system λ • . (3) G (0) is e quipp e d with a non-zer o R adon me asur e µ (0) which is quasi-invarian t with r esp e ct to λ • . Such a gr oup oid wil l b e c al le d a Haar gr oup oid . W e will denote a Haa r group oid by ( G, λ • , µ (0) ), or just by G when λ • and µ (0) are eviden t from the contex t. Definition 2.9. L et ( G, λ • , µ (0) ) and ( H, η • , ν (0) ) b e Haar gr oup oids. L et p : G → H b e a c on tinuous gr oup oid homomorphis m w hich is also me asur e cla ss pr eserving w i th r esp e ct to the induc e d me asur es, i.e. p ∗ ( µ ) ∼ ν . We say that p is a homomorphism of H aar gr oup oids . In the a b ov e definition p ∗ is the push-forw ard, defined fo r an y Borel set E ⊂ H by p ∗ µ ( E ) = µ ( p − 1 ( E )). A homomor phism of Haar g r o up o ids is also measure class preserving on t he unit spaces, as we shall shortly see. W e first need the follow ing fact. Lemma 2.10. L et ( G, λ • , µ (0) ) b e a Haar gr oup oid. The r ang e ma p r : G → G (0) satisfies r ∗ ( µ ) ∼ µ (0) . Pr o of. Let E ⊆ G (0) b e a Borel subset. W e need to show that µ ( r − 1 ( E )) = 0 if and only if µ (0) ( E ) = 0. By the definition of the induced measure, µ ( r − 1 ( E )) = R G (0) λ u ( r − 1 ( E )) dµ (0) ( u ) = R G (0) χ E ( u ) λ u ( G ) dµ (0) ( u ), since λ u ( r − 1 ( E )) = 0 if u / ∈ E whereas λ u ( r − 1 ( E )) = λ u ( G ) if u ∈ E . Since λ • is a Haar system, su p p ( λ u ) = G u 6 = ∅ , a nd in particular λ u ( G ) > 0 fo r ev ery u . It follows that µ ( r − 1 ( E )) = 0 if and o nly if χ E ( u ) = 0 µ (0) -a.e., whic h is if and only if µ (0) ( E ) = 0.  While the pro of we included ab o v e is elemen tary , w e p oint out that Lemma 2.10 also follows from the fa ct tha t b y the definition of the induced measure µ , the Haar system λ • is a 6 A. CENS OR A ND D. GRA NDINI disin tegration of µ with respect to µ (0) , whic h implies that r : G → G (0) is measure class preserving. See Lemma 6.4 of [4]. Sligh tly abusing notation, w e also denote the restriction of p to G (0) b y p . Prop osition 2.11. L et ( G, λ • , µ (0) ) and ( H , η • , ν (0) ) b e Haar g r oup oids, and let p : G → H b e a homom orphism of Haar gr oup oids. Then p ∗ ( µ (0) ) ∼ ν (0) . Pr o of. Consider the follow ing commu ting diagram: G p   r G / / G (0) p   H r H / / H (0) Let E ⊆ H (0) b e a Borel subset. W e need to sho w that µ (0) ( p − 1 ( E )) = 0 if and only if ν (0) ( E ) = 0. Indeed, b y Lemma 2.10 applied to H , ν (0) ( E ) = 0 ⇔ ν ( r − 1 H ( E )) = 0 ⇔ µ ( p − 1 ( r − 1 H ( E ))) = 0. A t the same time, b y Lemma 2.10 applied to G , w e hav e that µ (0) ( p − 1 ( E )) = 0 ⇔ µ ( r − 1 G ( p − 1 ( E ))) = 0. Since the diagram comm utes, p − 1 ( r − 1 H ( E )) = r − 1 G ( p − 1 ( E )), and it follows that ν (0) ( E ) = 0 ⇔ µ (0) ( p − 1 ( E )) = 0.  Ha ving defined Haa r group oids and their appropriate maps, we are ready to define the setting for this pap er and its sequels. Definition 2.12. We intr o duc e the c a te gory H G , which has Haar gr oup oids as obje cts and homomorphism s of Haar gr oup oids as morphis m s. 3. The top ological weak pullback The purp ose o f this pap er is t o construct a nd study the we ak pullbac k of Haar group oids. W e start b y constructing the w eak pullback of top olo gic al group oids. W e shall lea ve it to the reader to v erify tha t in the case o f discr ete group oids, our no tion of weak pullbac k reduces to the o ne in [2], whic h in turn generalizes the more familiar no tion of pullback in t he cat ego ry of sets. E xamples 3.4 and 3.5 b elow illustrat e that the w eak pullbac k is a natural not ion. Definition 3.1. Given the fol lowing diagr am of top olo gic al gr oup oids a nd c ontinuous homo- morphisms S p   ? ? ? ? ? ? ? T q   ~ ~ ~ ~ ~ ~ ~ G we define the we ak pul l b ack to b e the top olo gic al g r oup oid P = { ( s, g , t ) | s ∈ S, g ∈ G, t ∈ T , r G ( g ) = r G ( p ( s )) and d G ( g ) = r G ( q ( t )) } to gether w ith the obvious p r oje ctions π S : P → S and π T : P → T . We describ e the gr oup oid structur e of P and its top olo gy b elow. WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 7 The w eak pullbac k gro upo id P give s rise to the follow ing diagra m: P π S          π T   @ @ @ @ @ @ @ S p   ? ? ? ? ? ? ? T q   ~ ~ ~ ~ ~ ~ ~ G Observ e that ev en at the lev el of sets, this diagram do es n o t commute . Ho wev er, it is not hard to see that the weak pullbac k do es make t he f o llo wing diamond comm ute: P π S          π T   @ @ @ @ @ @ @ S π ◦ p   > > > > > > > T π ◦ q          G where π : G − → G is the map g 7− → [ r ( g )] = [ d ( g )]. In tuitive ly , w e think o f an elemen t ( s, g , t ) in P as giving rise to the following picture in G : p ( s )   q ( t )   g o o Comp osition of ( s, g , t ) a nd ( σ, h, τ ) is then tho ug h t o f as: p ( σ )   q ( τ )   p ( s )   h o o q ( t )   g o o F ormally , the comp osable pairs of P ar e P (2) = { ( s, g , t ) , ( σ , h, τ ) | r S ( σ ) = d S ( s ) , r T ( τ ) = d T ( t ) and h = p ( s ) − 1 g q ( t ) } . The pro duct is g iven b y ( s, g , t )( σ, h, τ ) = ( sσ, g , tτ ) , and the inv erse is giv en b y ( s, g , t ) − 1 = ( s − 1 , p ( s ) − 1 g q ( t ) , t − 1 ) . Th us the range and source ma ps of P are r P ( s, g , t ) = ( r S ( s ) , g , r T ( t )) 8 A. CENS OR A ND D. GRA NDINI and d P ( s, g , t ) = ( d S ( s ) , p ( s ) − 1 g q ( t ) , d T ( t )) . The unit space of P is P (0) = { ( s, g , t ) | s ∈ S (0) , t ∈ T (0) and g ∈ G p ( s ) q ( t ) } . The top olo g y of P is induced from the Cartesian pro duct S × G × T : X ⊆ P is op en ⇔ there exists an op en set Z ⊆ S × G × T suc h that X = Z ∩ P . The pro duct and in vers e of P are con tinuous with resp ect t o this top ology . Remark 3.2. L et { A n } ∞ n =1 , { B m } ∞ m =1 and { C k } ∞ k =1 b e c ountable b ases fo r the top olo gies o f S , G and T r esp e ctively. Then B = { ( A n × B m × C k ) ∩ P } ∞ n,m,k =1 gives a c ountable b as i s B for the top olo gy of P , c onsi s ting of op en sets of the form E = ( A × B × C ) ∩ P , which we c a l l elemen tary op en sets . Mor e over, al l finite interse ctions of sets in B ar e also of the this form. Lemma 3.3. The gr oup oid P is lo c al ly c omp act, Hausdorff a nd se c ond c o untable . Pr o of. The group oid P is second coun table by Remark 3.2, a nd it is Hausdorff a s a subspace of S × G × T . Let b : S × G × T − → G (0) × G (0) × G (0) × G (0) b e the contin uous map giv en by ( σ , x, τ ) 7− → ( r G ( p ( σ )) , r G ( x ) , d G ( x ) , r G ( q ( τ ))) . Observ e that P = b − 1 (∆ × ∆), where ∆ is the diagonal of G (0) × G (0) . Therefore, P is closed in S × G × T , and therefore it is lo cally compact.  The following examples show that the weak pullbac k of group oids is a natural notion. A more detailed study of these examples and man y others will app ear in a separate pap er, where w e discuss the w eak pullbac k in the conte xt of top ological and measure theoretic degroup oidification. Example 3.4. (w eak pullback o f op en co ver group oids) Let X , Y and Z b e lo cally compact top olo g ical spaces, and let p : Y → X and q : Z → X b e contin uous, op en and surjectiv e maps. Assum e that U = { U α } α ∈ A and W = { W α } α ∈ A are lo cally finite op en co vers of Y and Z , resp ectiv ely (with the same indexing set A ), and assume that p ( U α ) = q ( W α ) for eve ry α ∈ A , defining an open cov er V = { V α } α ∈ A of X , where V α = p ( U α ). Consider t he regular pullback diagram in the category T op of top ological spaces and contin uous functions: Y ∗ Z π Y | | y y y y y y y y π Z " " E E E E E E E E Y p " " E E E E E E E E E Z q | | y y y y y y y y y X where Y ∗ Z = { ( y , z ) ∈ Y × Z | p ( y ) = q ( z ) } . All sets of the form ( U α × W β ) ∩ Y ∗ Z constitute an op en cov er of the pullbac k space Y ∗ Z , whic h we will denote by U ∗ W . WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 9 Asso ciated t o an op en cov er U of a space Y is a group oid G U = { ( α, y , β ) : y ∈ U α ∩ U β } (called an o p en co v er group oid, or ˇ C ech group oid). A pair ( α , y , β ), ( γ , y ′ , δ ) is comp osable if and o nly if β = γ and y = y ′ , in whic h case their pr o duct is ( α , y , δ ), a nd t he in ve rse is giv en b y ( α , y , β ) − 1 = ( β , y , α ) . Let G U , G W and G V b e the op en co ver group oids asso ciated to the cov ers of Y , Z and X ab ov e, and let b p : G U → G V and b q : G W → G V b e the induced homomorphisms, given b y b p ( α, y , β ) = ( α , p ( y ) , β ) and b q ( α, z , β ) = ( α, q ( z ) , β ). This gives rise to a cospan diagra m o f gr o up o ids, which can b e completed to a w eak pullbac k diagr a m: P } } | | | | | | | | ! ! C C C C C C C C G U b p B B B B B B B B G W b q } } { { { { { { { { G V W e omit the tec hnical but straigh tforward calculatio ns whic h yield the upshot: the we ak pul lb ac k gr oup oid P is isomorphic to the op en co ver g r oup oid G U ∗W corresp onding t o the co ve r U ∗ W of the r e gular pul lb ack sp ac e Y ∗ Z . Example 3.5. (w eak pullback o f tra nsformation group oids) Let X , Y and Z b e lo cally compact top olo g ical spaces, and let p : Y → X and q : Z → X b e con tin uous maps. Let Y ∗ Z b e the regular pullbac k in the category T op , as in the previous example. Let Γ and Λ b e lo cally compact groups acting on Y and Z resp ectiv ely , and let Y × Γ a nd Z × Λ b e the corresp onding transformation group oids. Recall that in a transformation group o id, sa y Y × Γ, the elemen ts ( y , γ ) and ( ˜ y , ˜ γ ) a r e comp osable if and only if ˜ y = y γ , in whic h case ( y , γ )( y γ , ˜ γ ) = ( y , γ ˜ γ ). The inv erse, range and domain a re give n b y ( y , γ ) − 1 = ( y γ , γ − 1 ), r ( y , γ ) = ( y , e ) and d ( y , γ ) = ( y γ , e ). W e view X as a transformation g r oup oid b y endo wing it with an action of the trivial group, whic h amoun ts to regarding X as a cot r ivial group oid. Ass ume that t he maps p and q a re equiv ariant with resp ect to the gro up actions, i.e. p ( y · γ ) = p ( y ) a nd q ( z · λ ) = q ( z ). In this case p and q induce group oid homomorphisms ˆ p : Y × Γ → X and ˆ q : Z × Λ → X giv en b y ˆ p ( y , γ ) = p ( y ) a nd ˆ q ( z , λ ) = q ( z ). This yields a cospan diag ram of top olog ical group oids whic h giv es rise to the fo llo wing weak pullback diagram: P π Y | | x x x x x x x x x π Z " " F F F F F F F F F Y × Γ ˆ p " " F F F F F F F F Z × Λ ˆ q | | x x x x x x x x X It is now no t hard to ve rify that t he we ak pul lb ack gr oup oid P can be iden tified with the transformation group oid ( Y ∗ Z ) × (Γ × Λ ) corresp onding to the action of the gro up (Γ × Λ) o n the r e gular pul lb ack sp ac e ( Y ∗ Z ), giv en b y ( y , z ) · ( γ , λ ) = ( y γ , z λ ). Remark 3.6. In ge n er al, the we ak pul lb ack c oi n cides w ith a r e gular pul lb ack wh e never the gr o up oid G in Definition 3.1 is a c otrivial gr oup oid. This is the c ase in ex a mple 3.5 ab ove. 10 A. CENS OR A ND D. GRA NDINI The follow ing observ a tion will b e essen t ia l in the sequel. Lemma 3.7. F or an y u = ( s, g , t ) ∈ P (0) , the fib er P u is a c artesian pr o duct of the form P u = P ( s,g ,t ) = S s × { g } × T t . Pr o of. W e follo w the definitions: P ( s,g ,t ) = { ( σ , h, τ ) ∈ P | r P ( σ , h, τ ) = ( s, g , t ) } = { ( σ , h, τ ) ∈ P | ( r S ( σ ) , h, r T ( τ )) = ( s, g , t ) } = { ( σ , h, τ ) ∈ P | r S ( σ ) = s, h = g , r T ( τ ) = t } = { ( σ , h, τ ) ∈ P | σ ∈ S s , h = g , τ ∈ T t } . Note that sinc e ( s, g , t ) is an elemen t of P (0) , an y σ ∈ S s satisfies r G ( p ( σ )) = p ( r S ( σ )) = p ( s ) = p ( r S ( s )) = r G ( p ( s )) = r G ( g ) and lik ewise an y τ ∈ T t satisfies r G ( q ( τ )) = d G ( g ). Therefore S s × { g } × T t ⊆ P a nd th us P ( s,g ,t ) = { ( σ, h, τ ) ∈ P | σ ∈ S s , h = g , τ ∈ T t } = S s × { g } × T t .  Prop osition 3.8. T h e pr o j e ction s π S : P → S and π T : P → T ar e c ontinuous gr oup o i d homomorphism s. Pr o of. The pro of is straigh tforw ar d. F or contin uit y , let A ⊆ S b e a n op en subset. Then π − 1 S ( A ) is op en in P since π − 1 S ( A ) = { ( s, g , t ) ∈ P | π S ( s, g , t ) ∈ A } = { ( s, g , t ) ∈ P | s ∈ A } = ( A × G × T ) ∩ P . Now take (( s, g , t ) , ( σ, h, τ )) ∈ P (2) . Then π S (( s, g , t )( σ , h, τ )) = π S ( sσ , g , tτ ) = sσ = π S ( s, g , t ) π S ( σ , h, τ ). Also, π S (( s, g , t ) − 1 ) = π S ( s − 1 , p ( s ) − 1 g q ( t ) , t − 1 ) = s − 1 = ( π S ( s, g , t )) − 1 . Th us π S is a group oid homomorphism. The pro of for π T is similar.  4. A Haar s ystem for the we ak pullback W e now assume that S , G and T are Ha a r group oids and that the maps p and q are homomorphisms of Haar group oids. In order to define the w eak pullback of the following diagram in the category HG , w e let P b e t he w eak pullback o f the underlying diagram of top ological group oids, as defined ab ov e. P             > > > > > > > > λ • S , µ (0) S S p   = = = = = = = = T q           λ • T , µ (0) T G λ • G , µ (0) G Our goal is to construct a Haar group oid structure on P . W e start b y defining the Haar system λ • P . F rom Lemma 3 .7 w e kno w that the r -fib ers o f P are cartesian pro ducts of the form P u = P ( s,g ,t ) = S s × { g } × T t . In ligh t of this it is reasonable to prop ose the fo llo wing definition. Definition 4.1. L et u = ( s, g , t ) ∈ P (0) . Define λ u P = λ ( s,g ,t ) P := λ s S × δ g × λ t T . We denote λ • P = { λ u P } u ∈ P (0) . WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 11 Theorem 4.2. The system λ • P is a c ontinuous left Haar system for P . Pr o of. The pro of will rely on t he tec hnology dev elop ed in [4]. W e consider the followin g three pullbac k diagrams in the category T op of top ological spaces and contin uous functions (i.e. w e temp orarily forg et the a lg ebraic structures of the gro upo ids in volv ed, and view them only as top ological spaces. Lik ewise all group oid homomorphisms are regarded only a s contin uous functions): Diagram A G (0) ∗ G (0)   / / G (0) v 7→ [ v ]   G (0) u 7→ [ u ] / / G Diagram B S (0) ∗ T (0)   / / T (0) t 7→ [ q ( t )]   S (0) s 7→ [ p ( s )] / / G Diagram C S ∗ T   / / T τ 7→ [ q ( r ( τ ))]   S σ 7→ [ p ( r ( σ ))] / / G Note that in order to ligh ten notat ion, w e denote the pullbac k ob ject, for example in D ia gram C, b y S ∗ T in place of S ∗ G T . By definition S ∗ T = S ∗ G T = { ( σ, τ ) ∈ S × T | [ p ( r ( σ ))] = [ q ( r ( τ ))] in G } and the maps to S and T are the ob vious pro jections. The t op ology of S ∗ T is the r estriction of the pro duct top ology on S × T . Using G (0) ∗ G (0) , S (0) ∗ T (0) and S ∗ T , we can now construct tw o more pullback dia g rams (still in T op ). Our identifications o f the pullbac k ob jects in Diag r a ms D and E with P (0) and P , r esp ectiv ely , are justified b elow . A momen t’s reflection rev eals that the maps in these diagrams are we ll defined. Diagram D P (0)   / / S (0) ∗ T (0) ( s,t ) 7→ ( p ( s ) ,q ( t ))   G x 7→ ( r ( x ) ,d ( x )) / / G (0) ∗ G (0) 12 A. CENS OR A ND D. GRA NDINI Diagram E P   / / S ∗ T ( σ , τ ) 7→ ( p ( r ( σ )) ,q ( r ( τ )))   G x 7→ ( r ( x ) ,d ( x )) / / G (0) ∗ G (0) In D iagram D we iden t ified the pullback o b ject G ∗ G (0) ∗ G G (0) ( S (0) ∗ G T (0) ) with P (0) . Indeed, G ∗ G (0) ∗ G G (0) ( S (0) ∗ G T (0) ) = { ( g , ( s, t )) | ( r G ( g ) , d G ( g )) = ( p ( s ) , q ( t )) } = { ( g , ( s, t )) | r G ( g ) = p ( s ) a nd d G ( g ) = q ( t ) } = { ( g , ( s, t )) | g ∈ G p ( s ) q ( t ) } whic h can obvious ly b e iden tified, as sets, with our definition of P (0) . Moreo v er, the top ology on the pullback is precisely that o f P (0) , namely the induced top ology from S (0) × G × T (0) . Similarly , in Diagram E we identifie d the pullback ob ject G ∗ G (0) ∗ G G (0) ( S ∗ G T ) with P . Indeed, G ∗ G (0) ∗ G G (0) ( S ∗ G T ) = { ( g , ( s , t )) | ( r G ( g ) , d G ( g )) = ( p ( r S ( s )) , q ( r T ( t ))) } = { ( g , ( s, t )) | r G ( g ) = p ( r S ( s )) and d G ( g ) = q ( r T ( t )) } whic h can b e iden tified with our definition of P , as sets as w ell as in T op . Henceforth, we shall follow Section 5 of [4 ], where w e studied fibred pro ducts of systems of measures. Observ e tha t the results w e in vok e at this p oin t from [4] only require spaces to b e T 1 and second coun table. The spaces w e consider all satisfy these hy p otheses. Using Diagram C as the fron t face and Diagram B as the bac k face, w e construct the follow ing fibred pro duct diagr am: S (0) ∗ T (0) T (0) S (0) G S ∗ T T S G / / / / / / [ p ] / / [ p ◦ r S ]     [ q ]     [ q ◦ r T ]             ? ? r S ∗ r T             ? ? r T λ • T             ? ? r S λ • S             ? ? The connecting ma ps ar e the range maps r T and r S , and they are endo wed resp ectiv ely with the Haar systems λ • T and λ • S , whic h are con t inuous systems of measures and therefore lo cally finite (see Corollary 2.15 o f [4]). It is immediate t o see that the compatibility conditions on the maps of t he b ottom a nd the righ t faces are satisfied. The map r S ∗ r T : S ∗ T → S (0) ∗ T (0) WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 13 is defined by ( r S ∗ r T )( s, t ) = ( r S ( s ) , r T ( t )). By Definition 5.1 and Prop osition 5 .2 of [4], w e obtain a lo cally finite system of measures ( λ S ∗ λ T ) • on r S ∗ r T , where ( λ S ∗ λ T ) ( s,t ) = λ s S × λ t T . Moreo ve r, b y Prop osition 5.5 of [4] it is p o sitive on op en sets. With this at hand, w e construct anot her fibred pro duct diagram. W e tak e Diagr a m E as the fron t face and Diagram D as the bac k face, and use r S ∗ r T and id : G → G a s the connecting maps. The map r S ∗ r T is equipp ed with the ab o v e lo cally finite system of measures ( λ S ∗ λ T ) • , whereas the identit y map on G naturally admits the system δ • of Dirac masses, whic h is trivially lo cally finite: P (0) S (0) ∗ T (0) G G (0) ∗ G (0) P S ∗ T G G (0) ∗ G (0) / / / / ( r ,d ) / / / / ( r ,d )     p ∗ q     ( p ◦ r ) ∗ ( q ◦ r )             ? ? r P             ? ? r S ∗ r T ( λ S ∗ λ T ) •             ? ? id δ •             ? ? It is again easy to see that the compatibilit y conditions on the maps o f the b otto m a nd the righ t f aces are satisfied. Note that in this last diagram w e hav e iden tified the map from P to P (0) with r P , the ra nge map of P . Resorting once again to D efinition 5.1 and Prop osition 5.2 of [4], we obtain a lo cally finite system of measures ( δ ∗ ( λ S ∗ λ T )) • on r P : P → P (0) , where ( δ ∗ ( λ S ∗ λ T )) ( g, s,t ) = δ g × ( λ S ∗ λ T ) ( s,t ) = δ g × λ s S × λ t T . W e de note this system of measures on r P b y λ • P . Yielding to the original con v ention of writing elemen ts of P as ( s, g , t ) r a ther than ( g , s, t ), we write λ ( s,g ,t ) P = λ s S × δ g × λ t T . Our construction of λ • P as a fibred pro duct of the systems δ • and ( λ S ∗ λ T ) • , whic h are lo cally finite and p ositiv e on op en sets, guaran tees (by Prop ositions 5.2 and 5.5 of [4]) that λ • P inherits these prop erties. Recall t hat as w e ha v e p ointed out in the preliminaries, G need not b e a Hausdorff space in general. M oreo ver, S ∗ T , for example, need not b e lo cally compact, as it is not necessarily closed in S × T . The assumption that all spaces are lo cally compact and Hausdorff is essen tial in the CSM setting in [4]. F o r this reason w e cannot simply use Prop osition 5.4 of [4 ] t o deduce t ha t as fibred pro ducts, ( λ S ∗ λ T ) • and subsequen tly λ • P are CSMs. Thus , w e presen t a separate direct pro of that λ • P is a CSM in Prop osition 4.3 below. F urthermore, a t this p oin t we return to viewing P , G , S and T as g roup oids, and in Prop o sition 4.4 w e state and pro ve that λ • P is left inv arian t. W e conclude that λ • P is a con tinuous left Haa r system for the group oid P .  Prop osition 4.3. The system λ • P is a c ontinuous system of me asur es. 14 A. CENS OR A ND D. GRA NDINI Pr o of. F r o m the definition o f a CSM, in order t o prov e that λ • P is a CSM on r P : P → P (0) , w e need to show that f or a n y 0 ≤ f ∈ C c ( P ), the map ( s, g , t ) 7→ R P f ( σ, x, τ ) dλ ( s,g ,t ) P ( σ , x, τ ) is a con tin uous function on P (0) . Let 0 ≤ f ∈ C c ( P ). Recall f rom the pro o f of Lemma 3.3 that P is closed in S × G × T . By Tietze’s Extension Theorem, there exists a function F ∈ C ( S × G × T ) suc h that F | P = f . Since w e can m ultiply F by a function ϕ ∈ C c ( S × G × T ) whic h satisfies ϕ = 1 on K = supp ( f ), w e can assume, without loss o f generality , that F ∈ C c ( S × G × T ). W e no w resort to (symmetric v ersions of ) Lemma 4.5 in [4]. First w e tak e X = S × G , Y = T , Z = T (0) and γ • = λ • T , to deduce that the func tion F 1 defined by ( σ , x, t ) 7→ R T F ( σ , x, τ ) d λ t T ( τ ) is in C c ( S × G × T (0) ). Next, taking X = S × T (0) , Y = G , Z = G and γ • = δ • , w e get that the f unction F 2 defined b y ( σ, g , t ) 7→ R G F 1 ( σ , x, t ) dδ g ( x ) is in C c ( S × G × T (0) ). Finally , with X = G × T (0) , Y = S , Z = S (0) and γ • = λ • S , Lemma 4.5 of [4] implies that the function F 3 defined b y ( s, g , t ) 7→ R S F 2 ( σ , g , t ) d λ s S ( σ ) is in C c ( S (0) × G × T (0) ). Merging these results, w e can rewrite the function F 3 b y ( s, g , t ) 7− → Z S Z G Z T F ( σ , x, τ ) dλ t T ( τ ) dδ g ( x ) λ s S ( σ ) . Note that in the ab ov e in tegral r S ( σ ) = s and r T ( τ ) = t , since supp ( λ s S ) = r − 1 S ( s ) a nd supp ( λ t T ) = r − 1 T ( t ). Therefore, if w e tak e ( s , g , t ) ∈ P (0) , in whic h case p ( s ) = r G ( g ) and q ( t ) = d G ( g ), w e get that p ( r S ( σ )) = r G ( g ) and q ( r T ( τ )) = d G ( g ). In o t her w ords, w hen restricting F 3 to P (0) , w e are actually in tegrating o v er P . Recalling the definition of λ • P and that F | P = f , we retrieve precisely t he function ( s, g , t ) 7→ R P f ( σ, x, τ ) dλ ( s,g ,t ) P ( σ , x, τ ) , whic h is contin uo us on P (0) as a restriction of a contin uous function on S (0) × G × T (0) .  Prop osition 4.4. The system λ • P is left invaria n t. Pr o of. F r o m the definition of left inv ar ia nce, we need to show that (3) λ d P ( x ) P ( E ) = λ r P ( x ) P  x · ( E ∩ P d P ( x ) )  , for ev ery x ∈ P and f o r ev ery Borel subset E ⊆ P . Assume first tha t E is a set of the form E = ( A × B × C ) ∩ P , where A ⊆ S , B ⊆ G and C ⊆ T . Let x = ( σ, y , τ ) ∈ P , so r P ( x ) = ( r S ( σ ) , y , r T ( τ )) and d P ( x ) = ( d S ( σ ) , p ( σ ) − 1 y q ( τ ) , d T ( τ )). W e will denote z = p ( σ ) − 1 y q ( τ ). W e calculate the left and right hand sides of (3) separately . On the o ne hand we get: λ d P ( x ) P ( E ) = λ d P ( x ) P  ( A × B × C ) ∩ P d P ( x )  since λ d P ( x ) P is concen trated on P d P ( x ) = λ d P ( x ) P  ( A × B × C ) ∩ ( S d S ( σ ) × { z } × T d T ( τ ) )  b y Lemma 3 .7 = λ d P ( x ) P  ( A ∩ S d S ( σ ) ) × ( B ∩ { z } ) × ( C ∩ T d T ( τ ) )  = λ d S ( σ ) S ( A ∩ S d S ( σ ) ) · δ z ( B ∩ { z } ) · λ d T ( τ ) T ( C ∩ T d T ( τ ) ) = λ d S ( σ ) S ( A ) · δ z ( B ) · λ d T ( τ ) T ( C ) On the o ther hand, λ r P ( x ) P  x · ( E ∩ P d P ( x ) )  = λ r P ( x ) P  ( σ , y , τ ) ·  ( A × B × C ) ∩ P d P ( x )  = λ r P ( x ) P  ( σ , y , τ ) ·  ( A ∩ S d S ( σ ) ) × ( B ∩ { z } ) × ( C ∩ T d T ( τ ) )  WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 15 By the definition of P (2) , note tha t ( σ , y , τ ) ·  ( A ∩ S d S ( σ ) ) × ( B ∩ { z } ) × ( C ∩ T d T ( τ ) )  can b e nonempty only when z = p ( σ ) − 1 y q ( τ ) ∈ B , in whic h case the middle comp onen t of the pro duct is { y } . Hence = ( λ r P ( x ) P  σ · ( A ∩ S d S ( σ ) ) × { y } × τ · ( C ∩ T d T ( τ ) )  z ∈ B λ r P ( x ) P ( ∅ ) z / ∈ B = ( λ r S ( σ ) S  σ · ( A ∩ S d S ( σ ) )  · δ y ( { y } ) · λ r T ( τ ) T  τ · ( C ∩ T d T ( τ ) )  z ∈ B 0 z / ∈ B = ( λ d S ( σ ) S ( A ) · λ d T ( τ ) T ( C ) z ∈ B 0 z / ∈ B b y the left inv ariance of λ • S and λ • T = λ d S ( σ ) S ( A ) · δ z ( B ) · λ d T ( τ ) T ( C ) Th us (3) holds for any set E of the f orm E = ( A × B × C ) ∩ P . Fix x ∈ P , and for any Borel subset E of P define µ ( E ) = λ d P ( x ) P ( E ) and ν ( E ) = λ r P ( x ) P  x · ( E ∩ P d P ( x ) )  . W e claim t ha t µ a nd ν are both lo cally finite measures on P . Since λ • P is a CSM, it is a lo cally finite BSM b y Prop osition 2.23 of [4]. Hence λ u P is a lo cally finite measure for a ny u ∈ P (0) , and in particular µ = λ d P ( x ) P is a lo cally finite measure. W e turn to ν . It is trivial that ν ( ∅ ) = 0 . Let { E i } ∞ i =1 b e a coun table collection of disjoin t Borel subsets of P . ν ( ∞ [ i =1 E i ) = λ r P ( x ) P ( x · ( ( ∞ [ i =1 E i ) ∩ P d P ( x ) )) = λ r P ( x ) P ( x · ( ∞ [ i =1 ( E i ∩ P d P ( x ) ))) = = λ r P ( x ) P ( ∞ [ i =1 x · ( E i ∩ P d P ( x ) )) = ∞ X i =1 λ r P ( x ) P  x ·  E i ∩ P d P ( x )  = ∞ X i =1 ν ( E i ) . Therefore ν is countably additiv e, and henc e a measure. In order to pro v e that ν is lo- cally finite w e need to sho w tha t ev ery y ∈ P admits a n op en neigh b orho o d U y suc h that ν ( U y ) < ∞ . In the case where y / ∈ P d P ( x ) , the open set U y = P \ P d P ( x ) satisfies ν ( U y ) = λ r P ( x ) P  x · ( U y ∩ P d P ( x ) )  = λ r P ( x ) P ( ∅ ) = 0 < ∞ . Now assume t ha t y ∈ P d P ( x ) . In this case the pro duct z = xy is well defined, and since λ r P ( x ) P is a lo cally finite measure, there exists an op en neighborho o d U z of z suc h that λ r P ( x ) P ( U z ) < ∞ . The map P d P ( x ) → P de- fined b y w 7→ x · w is con tin uous, hence there exists an op en neigh b orho o d U y of y suc h that x ·  U y ∩ P d P ( x )  ⊂ U z . Consequen tly , ν ( U y ) = λ r P ( x ) P  x · ( U y ∩ P d P ( x ) )  ≤ λ r P ( x ) P ( U z ) < ∞ . Finally , let B be a countable basis for the top o logy of P consisting of elemen tary op en sets, as in R emark 3.2 . As w e ha v e just shown, elemen tary op en sets satisfy (3), henc e µ and ν ag r ee on all finite interse ctions of sets in B . W e can now inv ok e Lemma 2.24 of [4 ], whic h states that if µ and ν are t w o lo cally finite measures on a space X , and there exists a coun table basis B for the top ology of X suc h that µ ( U 1 ∩ U 2 ∩· · ·∩ U n ) = ν ( U 1 ∩ U 2 ∩· · ·∩ U n ) for an y { U 1 , U 2 , . . . , U n } ⊂ B , n ≥ 1 , then µ ( E ) = ν ( E ) f or any Borel subset E ⊆ X . Apply ing Lemma 2.24 of [4] to µ , ν a nd B ab o v e completes the pro of.  16 A. CENS OR A ND D. GRA NDINI 5. A meas ure on the unit sp ace of t he weak pullback W e return to t he we ak pullback diagr a m. Our next task is to construct a measu re µ (0) P on P (0) , and for start ers we will need to hav e certain systems of measures γ • p and γ • q on the maps p and q , resp ectiv ely . Thes e systems o f measures ar ise via a disin tegration theorem, as w e explain b elow. P             > > > > > > > > λ • P λ • S , µ (0) S S p,γ • p   = = = = = = = = T q ,γ • q           λ • T , µ (0) T G λ • G , µ (0) G Let ( X , µ ) and ( Y , ν ) b e measure spaces, and let f : X → Y b e a Bo r el map. A system of measures γ • on f will b e called a disi n te gr ation ([4 ], Definition 6 .2) of µ with resp ect to ν if µ ( E ) = Z Y γ y ( E ) dν ( y ) for ev ery Borel set E ⊆ X . A disinte gr ation the or em giv es sufficien t conditions whic h guarantee the existence of suc h a disin t egra tion, and the v ersion w e will use app ears as Corollary 6.6 of [4]. It requires µ to b e lo cally finite (and σ -finite), ν to b e σ -finite, and f : X → Y to b e measure class preserving. Under these conditions there exists a lo cally finite BSM γ • on f whic h is a disinte gration of µ with resp ect to ν . Eac h of the Haar group oids S , G and T is equipp ed with a Rado n (hence lo cally finite and σ -finite) measure on its unit spaces, whic h is quasi-in v ariant with resp ect to its Haar system. The maps p and q are homomorphisms of Haar group oids, there fore p : S (0) → G (0) and q : T (0) → G (0) are measure class preserving. These ingredien ts allow us to in vok e Corollary 6.6 of [4], and to obtain lo cally finite BSMs γ • p on p : S (0) → G (0) whic h is a disin tegration of µ (0) S with resp ect to µ (0) G , a nd γ • q on q : T (0) → G (0) whic h is a disin tegratio n of µ (0) T with resp ect to µ (0) G . The follo wing r equiremen t will b e essen tial for our pro of of Prop osition 5.6 b elow, whic h states that t he measure µ (0) P whic h w e are constructing is lo cally finite. Assumption 5.1. We wil l hen c eforth assume that the disinte gr ation systems γ • p and γ • q c a n b e taken to b e lo c al ly b ounde d. Remark 5.2. By L emm a 2.11 of [4] , a CSM is always lo c al ly b ounde d. Ther efor e, a n appr opriate disinte gr ation the or em that pr o d uc es a system whic h is either a CSM or at le as t lo c al ly b ounde d would have al lowe d us to r emove Assumption 5.1. Con tinuous (hence lo cally b o unded) disin tegrations are abundant: Ex amples include dis- in tegrat ions of Leb esgue measures along maps from R n to R m , as w ell as fib er bundles t ha t admit a con tin uous disin tegratio n of a measure o n the total space with resp ect to a mea- sure on the base space. Seda show s that more general constructions of fib er spaces also host contin uous disin tegratio ns, see Theorem 3.2 o f [ 1 2]. In our conte xt, a Haar system is of course a contin uous disin tegration of the induced measure with respect to the measure on the unit space. A v ery general result (see Theorem 5.43 of [7], whic h is a cor o llary of Theorem 3.3 of [3]) states that an y con tinuous and op en map f : X → Y b etw een second coun table lo cally compact Hausdorff spaces, admits a con tinuous system of measures γ • . In WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 17 particular this implies that if ν is a measure on Y a nd w e d efine the measure µ on X via γ • b y µ ( E ) = Z Y γ y ( E ) dν ( y ), then γ • is a contin uous disin tegratio n o f µ with resp ect to ν . The next step is to construct a BSM on the pro jection π G : P (0) → G , using γ • p and γ • q . Prop osition 5.3. The pr o je ction π G : P (0) → G admits a lo c al ly finite BSM η • , given b y η x = γ r ( x ) p × δ x × γ d ( x ) q . Pr o of. W e form the following fibred pro duct diagram in the category T op , with Dia g ram B as the fro n t face and Diagram A as the ba c k face. The connecting maps a r e p : S (0) → G (0) and q : T (0) → G (0) , equip p ed with the lo cally finite BSMs γ • p and γ • q constructed ab ov e. The compatibility conditio ns on the maps of the b ottom and the righ t faces ar e easily seen to b e satisfied. G (0) ∗ G (0) G (0) G (0) G S (0) ∗ T (0) T (0) S (0) G / / / / / / / / [ p ]         [ q ]             ? ? p ∗ q             ? ? q γ • q             ? ? p γ • p             ? ? W e p oin t out that the results w e use from [4] throughout this pro o f do not require spaces to b e lo cally compact and Hausdorff. By Prop osition 5.2 in [4], w e obtain fro m t he ab o v e diagram the lo cally finite BSM ( γ p ∗ γ q ) • on p ∗ q : S (0) ∗ T (0) → G (0) ∗ G (0) , where ( γ p ∗ γ q ) ( u,v ) = γ u p × γ v q . Next, w e consider the follo wing pullbac k diagram in T op (this w as Diagram D in the pro of of Theorem 4.2). W e equip the map p ∗ q with the BSM ( γ p ∗ γ q ) • : P (0) π G   / / S (0) ∗ T (0) ( γ p ∗ γ q ) • p ∗ q   G ( r ,d ) / / G (0) ∗ G (0) W e follo w Section 4 of [4], wh ere w e studied lifting of systems of measures. By Definition 4.1, Remark 4.2 a nd Prop o sition 4.4 of [4], we can lift the lo cally finite BSM ( γ p ∗ γ q ) • and obtain a lo cally finite BSM (( r , d ) ∗ ( γ p ∗ γ q )) • on the pro jection π G : P (0) → G . W e denote η • = (( r, d ) ∗ ( γ p ∗ γ q )) • , and from the definition of lifting it follo ws tha t fo r x ∈ G , η x = δ x × ( γ p ∗ γ q ) ( r ( x ) ,d ( x )) = δ x × γ r ( x ) p × γ d ( x ) q , whic h we rewrite as η x = γ r ( x ) p × δ x × γ d ( x ) q . This completes the pro of.  18 A. CENS OR A ND D. GRA NDINI Lemma 5.4. L et E ⊆ P (0) b e a set of the form E = ( A × B × C ) ∩ P (0) , w her e A ⊆ S (0) , B ⊆ G a n d C ⊆ T (0) . F or any x ∈ G , η x ( E ) = γ r ( x ) p ( A ) δ x ( B ) γ d ( x ) q ( C ) . Pr o of. F r o m the definition of η • in Proposition 5.3 ab ov e, w e ha v e that η x ( E ) = ( γ r ( x ) p × δ x × γ d ( x ) q )  ( A × B × C ) ∩ P (0)  . Clearly if x / ∈ B then η x ( E ) = 0. If x ∈ B then, since δ x is concen trated on { x } , w e can write η x ( E ) = ( γ r ( x ) p × δ x × γ d ( x ) q )  ( A × { x } × C ) ∩ P (0)  . A p oint ( s, x, t ) ∈ P (0) whose G comp onent is x , satisfies s ∈ p − 1 ( r ( x )) and t ∈ q − 1 ( d ( x )), hence for x ∈ B we ha v e η x ( E ) = γ r ( x ) p ( A ∩ p − 1 ( r ( x ))) · δ x ( { x } ) · γ d ( x ) q ( C ∩ q − 1 ( d ( x ))) . Since supp ( γ r ( x ) p ) = p − 1 ( r ( x )) and supp ( γ d ( x ) q ) = q − 1 ( d ( x )), it follows that for x ∈ B , η x ( E ) = γ r ( x ) p ( A ) δ x ( { x } ) γ d ( x ) q ( C ) . W e conclude that for an y x ∈ G , η x ( E ) = γ r ( x ) p ( A ) δ x ( B ) γ d ( x ) q ( C ) .  W e can now co ok up a measure µ (0) P on P (0) . The ingredien ts will b e the induced measure µ G from D efinition 2.2, as w ell as η • whic h w e hav e just constructed. Definition 5.5. L et B ⊆ P (0) b e a Bor el subset. Define: µ (0) P ( B ) := Z G η x ( B ) dµ G ( x ) . In fact, the measure µ (0) P can b e written as µ (0) P = µ G ◦ [( r , d ) ∗ ( γ p ∗ γ q )] , as it w as obta ined by lifting the fibred pro duct of the disin tegra tions γ p and γ q to π G : P (0) → G and then comp osing with the induced measure of G . In order for P to b e a Haar group oid, µ (0) P m ust b e a R adon measure, and in particular lo cally finite. T his is guarante ed mo dulo o ur standing Assumption 5.1. Prop osition 5.6. µ (0) P is a R a don m e asur e on P (0) . Pr o of. It suffices to sho w that µ (0) P is lo cally finite. Let A ⊆ S (0) , B ⊆ G and C ⊆ T (0) b e op en subsets with compact closures and consider the set E = ( A × B × C ) ∩ P (0) , whic h is an op en subset o f P (0) . Using the definition o f µ (0) P ab o v e alo ng with Lemma 5 .4, we get µ (0) P ( E ) = Z G η x ( E ) dµ G ( x ) = Z G γ r ( x ) p ( A ) δ x ( B ) γ d ( x ) q ( C ) d µ G ( x ) = Z B γ r ( x ) p ( A ) γ d ( x ) q ( C ) d µ G ( x ) . It thus follows fr o m Assumption 5.1 that µ (0) P ( E ) ≤  sup s γ s p ( A )  ·  sup t γ t q ( C )  · µ G ( B ) < ∞ . Since the op en sets of t he same form as E constitute a basis for t he top o logy of P (0) , w e conclude that µ (0) P is lo cally finite.  Note t ha t an alternativ e pro of of Prop osition 5 .6 is obtained by arguing that t he system η • is lo cally b ounded ( mo dulo Assumption 5.1), and then a pplying Coro llary 3.7 of [4]. Prop osition 5.7. T h e me asur e µ (0) P is indep end ent of the choic e of the disinte gr ations γ • p and γ • q . WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 19 Pr o of. Let e γ p • and e γ q • b e t w o other disin tegratio ns on p a nd q resp ectiv ely , and let e µ (0) P b e the corresp onding measure on P (0) . By Corollary 6.6 in [4], e γ p u = γ u p and e γ q u = γ u q for µ (0) G -almost ev ery u in G (0) . Let A ⊆ S (0) , B ⊆ G and C ⊆ T (0) b e o p en and let E = ( A × B × C ) ∩ P (0) b e the corresp onding op en subset of P (0) . By the calculation in the pro of of Prop osition 5.6 ab ov e, µ (0) P ( E ) = R B γ r ( x ) p ( A ) γ d ( x ) q ( C ) d µ G ( x ) , and lik ewise e µ (0) P ( E ) = R B e γ p r ( x ) ( A ) e γ q d ( x ) ( C ) d µ G ( x ) . Using Lemma 2.4 and the f a ct that supp ( λ u G ) = r − 1 ( u ) w e get e µ (0) P ( E ) = Z B e γ p r ( x ) ( A ) e γ q d ( x ) ( C ) d µ G ( x ) = Z G (0)  Z B e γ p r ( x ) ( A ) e γ q d ( x ) ( C ) d λ u G ( x )  dµ (0) G ( u ) = Z G (0) e γ p u ( A )  Z B e γ q d ( x ) ( C ) d λ u G ( x )  dµ (0) G ( u ) = Z G (0) γ u p ( A )  Z B e γ q d ( x ) ( C ) d λ u G ( x )  dµ (0) G ( u ) = Z G (0)  Z B γ r ( x ) p ( A ) e γ q d ( x ) ( C ) d λ u G ( x )  dµ (0) G ( u ) Justification fo r the next step is based on f orm ula (2) of Remark 2.7. The remaining calcu- lation retraces t he previous ar g umen ts. = Z G (0)  Z B γ d ( x ) p ( A ) e γ q r ( x ) ( C )∆ − 1 G ( x ) dλ u G ( x )  dµ (0) G ( u ) = Z G (0) e γ q u ( C )  Z B γ d ( x ) p ( A )∆ − 1 G ( x ) dλ u G ( x )  dµ (0) G ( u ) = Z G (0) γ u q ( C )  Z B γ d ( x ) p ( A )∆ − 1 G ( x ) dλ u G ( x )  dµ (0) G ( u ) = Z G (0)  Z B γ d ( x ) p ( A ) γ r ( x ) q ( C )∆ − 1 G ( x ) dλ u G ( x )  dµ (0) G ( u ) = Z G (0)  Z B γ r ( x ) p ( A ) γ d ( x ) q ( C ) d λ u G ( x )  dµ (0) G ( u ) = µ (0) P ( E ) Th us, e µ (0) P ( E ) = µ (0) P ( E ) for an y op en set of the form E = ( A × B × C ) ∩ P (0) . Thes e sets constitute a coun table basis B (0) for the top ology of P (0) , in analogy to Remark 3.2. Therefore, since µ (0) P is lo cally finite, it follo ws that e µ (0) P is lo cally finite as w ell. Moreo v er, µ (0) P and e µ (0) P agree on finite in tersections of sets in B (0) as these sets are a lso in B (0) , so w e can now use Lemma 2.24 of [4], as in the pro of of Prop osition 4.4, and conclude that e µ (0) P = µ (0) P .  The follo wing is a simple observ at io n, whose pro of is analogous to the pro o f of Lemma 2.4, and thus omitted. Lemma 5.8. F or any Bor el function f on P (0) : Z P (0) f ( u ) dµ (0) P ( u ) = Z G  Z P (0) f ( u ) dη y ( u )  dµ G ( y ) . 20 A. CENS OR A ND D. GRA NDINI In § 3 of [4] we defined the comp osition ( β ◦ α ) • of BSMs X p α • / / Y q β • / / Z , whic h is ch aracterized b y (4) Z X f ( x ) d ( β ◦ α ) z ( x ) = Z Y  Z X f ( x ) dα y ( x )  dβ z ( y ) . This will b e essen tial for proving the following lemma. Lemma 5.9. F or any Bor el function f ( y , σ ) on G ∗ S , Z S (0) Z S Z G f ( y , σ ) dλ p ( r S ( σ )) G ( y ) d λ s S ( σ ) dγ u p ( s ) = Z G Z S (0) Z S f ( y , σ ) dλ s S ( σ ) dγ r G ( y ) p ( s ) dλ u G ( y ) . Pr o of. Consider the comp osition ( γ p ◦ λ S ) • of the BSMs S r S λ • S / / S (0) p γ • p / / G (0) . W e use this as the right edge in the pull-bac k diagram b elow. F ollo wing § 4 of [4], w e lift the BSM λ • G to obtain a BSM ( ( p ◦ r S ) ∗ λ G ) • on π S : G ∗ S → S , and w e lift t he BSM ( γ p ◦ λ S ) • to obtain a BSM ( r ∗ G ( γ p ◦ λ S )) • on π G : G ∗ S → G . G ∗ S π G ( r ∗ G ( γ p ◦ λ S )) •   π S (( p ◦ r S ) ∗ λ G ) • / / S p ◦ r S ( γ p ◦ λ S ) •   G r G λ • G / / G (0) By the definition of lifting , (( p ◦ r S ) ∗ λ G ) σ = λ p ( r S ( σ )) G × δ σ , σ ∈ S and ( r G ∗ ( γ p ◦ λ S )) y = δ y × ( γ p ◦ λ S ) r G ( y ) , y ∈ G. The ab ov e diagra m g iv es rise to t w o comp ositions: G ∗ S π S (( p ◦ r S ) ∗ λ G ) • / / S p ◦ r S ( γ p ◦ λ S ) • / / G (0) and G ∗ S π G ( r ∗ G ( γ p ◦ λ S )) • / / G r G λ • G / / G (0) . How ev er, prop osition 4.8 of [4] stat es that the ab ov e diagram is a comm uta tiv e diagram o f BSMs, and explicitly , [( γ p ◦ λ S ) ◦ (( p ◦ r S ) ∗ λ G )] • = [ λ G ◦ ( r G ∗ ( γ p ◦ λ S )] • , as BSMs on G ∗ S → G (0) . The a b o v e equalit y implies that f o r any Borel function f ( y , σ ) on G ∗ S , Z G ∗ S f ( y , σ ) d (( γ p ◦ λ S ) ◦ (( p ◦ r S ) ∗ λ G )) u ( y , σ ) = Z G ∗ S f ( y , σ ) d ( λ G ◦ ( r G ∗ ( γ p ◦ λ S )) u ( y , σ ) . WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 21 W e expand the left and the righ t hand sides of t he ab o ve equalit y separately , using r ep eatedly the c haracterization (4) of comp osition of BSMs a b ov e: LH S = Z G ∗ S f ( y , σ ) d (( γ p ◦ λ S ) ◦ (( p ◦ r S ) ∗ λ G )) u ( y , σ ) = Z S  Z G ∗ S f ( y , σ ) d (( p ◦ r S ) ∗ λ G ) e σ ( y , σ )  d ( γ p ◦ λ S ) u ( e σ ) = Z S (0) Z S  Z G ∗ S f ( y , σ ) d (( p ◦ r S ) ∗ λ G ) e σ ( y , σ )  dλ s S ( e σ ) d γ u p ( s ) = Z S (0) Z S  Z G ∗ S f ( y , σ ) d ( λ p ( r S ( e σ )) G × δ e σ )( y , σ )  dλ s S ( e σ ) d γ u p ( s ) = Z S (0) Z S  Z G ∗ S f ( y , σ ) dλ p ( r S ( e σ )) G ( y ) d δ e σ ( σ )  dλ s S ( e σ ) d γ u p ( s ) = Z S (0) Z S Z G f ( y , σ ) dλ p ( r S ( σ )) G ( y ) d λ s S ( σ ) dγ u p ( s ) RH S = Z G ∗ S f ( y , σ ) d ( λ G ◦ ( r G ∗ ( γ p ◦ λ S )) u ( y , σ ) = Z G  Z G ∗ S f ( y , σ ) d ( r G ∗ ( γ p ◦ λ S )) e y ( y , σ )  dλ u G ( e y ) = Z G  Z G ∗ S f ( y , σ ) d ( δ e y × ( γ p ◦ λ S ) r G ( e y ) )( y , σ )  dλ u G ( e y ) = Z G  Z G ∗ S f ( y , σ ) dδ e y ( y ) d ( γ p ◦ λ S ) r G ( e y ) ( σ )  dλ u G ( e y ) = Z G  Z S f ( y , σ ) d ( γ p ◦ λ S ) r G ( y ) ( σ )  dλ u G ( y ) = Z G Z S (0) Z S f ( y , σ ) dλ s S ( σ ) dγ r G ( y ) p ( s ) dλ u G ( y ) Since the a b ov e expressions are equal, this yields the desired fo rm ula.  Lemma 5.10. L et f ( σ, x, τ ) b e a Bor el function on P . Then Z P f ( σ, x, τ ) dµ P ( σ , x, τ ) = Z G (0) Z G Z S (0) Z S Z T (0) Z T f ( σ, y , τ ) d λ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) . Pr o of. Z P f ( σ, x, τ ) dµ P ( σ , x, τ ) = = Z P (0) Z P f ( σ, x, τ ) dλ ( s,g ,t ) P ( σ , x, τ ) d µ (0) P ( s, g , t ) (b y Lemma 2 .4) = Z G Z P (0) Z P f ( σ, x, τ ) dλ ( s,g ,t ) P ( σ , x, τ ) d η y ( s, g , t ) d µ G ( y ) (b y Lemma 5.8) 22 A. CENS OR A ND D. GRA NDINI Rewriting η y b y Prop osition 5.3, a nd then rewriting λ ( s,g ,t ) P b y Definition 4.1, w e get = Z G Z Z Z S (0) × G × T (0) Z P f ( σ, x, τ ) dλ ( s,g ,t ) P ( σ , x, τ ) d γ r ( y ) p ( s ) dδ y ( g ) dγ d ( y ) q ( t ) dµ G ( y ) = Z G Z Z Z S (0) × G × T (0) Z Z Z S × G × T f ( σ, x, τ ) dλ s S ( σ ) dδ g ( x ) dλ t T ( τ ) dγ r ( y ) p ( s ) dδ y ( g ) dγ d ( y ) q ( t ) dµ G ( y ) = Z G Z Z S (0) × T (0) Z Z S × T f ( σ, y , τ ) d λ s S ( σ ) dλ t T ( τ ) dγ r ( y ) p ( s ) dγ d ( y ) q ( t ) dµ G ( y ) Using Lemma 2.4 again, fo llo w ed b y F ubini’s theorem, w e ha v e = Z G (0) Z G Z Z S (0) × T (0) Z Z S × T f ( σ, y , τ ) d λ s S ( σ ) dλ t T ( τ ) dγ r ( y ) p ( s ) dγ d ( y ) q ( t ) dλ u G ( y ) d µ (0) G ( u ) = Z G (0) Z G Z S (0) Z T (0) Z S Z T f ( σ, y , τ ) d λ t T ( τ ) dλ s S ( σ ) dγ d ( y ) q ( t ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) W e now in vok e Proposition 5.6 from [4], whic h asserts that for lo cally finite BSMs, fibred pro ducts comm ute with comp ositions. W e apply this theorem to the fo llo wing diagra m (it is straigh t f orw ard to ve rify that the conditions fo r the prop osition indeed hold. In particular, λ • S and λ • T are lo cally b ounded). W e o bta in that ( γ q ∗ γ p ) ◦ ( λ T ∗ λ S ) = ( γ q ◦ λ T ) ∗ ( γ p ◦ λ S ). T ∗ S r T ∗ r S ( λ T ∗ λ S ) • / / } } z z z z z z z z z z z z z T (0) ∗ S (0) q ∗ p ( γ q ∗ γ p ) • / / | | y y y y y y y y y y y y G (0) ∗ G (0)   } } | | | | | | | | | | | S r S λ • S / /     S (0) p γ • p / /     G (0)   T r T λ • T } } z z z z z z z z z z z z z / / T (0) q γ • q | | y y y y y y y y y y y y / / G (0) ~ ~ | | | | | | | | | | | | G id / / G id / / G Therefore, returning to our main calculation, w e get = Z G (0) Z G Z S (0) Z S Z T (0) Z T f ( σ, y , τ ) d λ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) This completes the pro of.  Prop osition 5.11. The me asur e µ (0) P is quasi-inv ariant with r esp e ct to λ • P . Pr o of. By definition 2.6, we need to sho w that µ P and µ − 1 P are m utually absolutely con- tin uous. W e recall from Definition 2.2 that µ P is the induce d measure, defined for an y Borel set E ⊆ P b y µ P ( E ) = R P (0) λ v P ( E ) dµ (0) P ( v ), and µ − 1 P is its image under in v ersion, i.e. µ − 1 P ( E ) = µ P ( E − 1 ). W e will prov e: Claim : There exists a function Λ : P → R satisfying Λ( α ) > 0 µ P -a.e., such tha t fo r an y Borel set E ⊆ P , µ − 1 P ( E ) = Z P χ E ( α )Λ( α ) dµ P ( α ) . WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 23 It will then follow that µ P ∼ µ − 1 P , since µ P ( E ) = R P χ E ( α ) dµ P ( α ). In fact, ∆ = Λ − 1 will b e the mo dular function of µ P . W e first pro ve the claim for elemen tar y op en subsets of the form E = ( A × B × C ) ∩ P , where A ⊆ S , B ⊆ G and C ⊆ T . Note that the characteristic function χ E is the restriction of the pro duct χ A · χ B · χ C to P . W e denote α = ( σ, x, τ ) ∈ P and v = ( s , g , t ) ∈ P (0) . By Lemma 5.10: µ − 1 P ( E ) = µ P ( E − 1 ) = Z P χ E − 1 ( σ , x, τ ) d µ P ( σ , x, τ ) = Z G (0) Z G Z S (0) Z S Z T (0) Z T χ E − 1 ( σ , y , τ ) d λ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) = Z G (0) Z G Z S (0) Z S Z T (0) Z T χ E ( σ − 1 , p ( σ ) − 1 y q ( τ ) , τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) = Z G (0) Z G Z S (0) Z S Z T (0) Z T χ A ( σ − 1 ) χ B ( p ( σ ) − 1 y q ( τ )) χ C ( τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) Using Lemma 5.9, w e obtain = Z G (0) Z S (0) Z S Z G Z T (0) Z T χ A ( σ − 1 ) χ B ( p ( σ ) − 1 y q ( τ )) χ C ( τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ p ( r ( σ )) G ( y ) d λ s S ( σ ) dγ u p ( s ) dµ (0) G ( u ) = Z G (0) Z S (0) Z S χ A ( σ − 1 ) Z G Z T (0) Z T χ B ( p ( σ ) − 1 y q ( τ )) χ C ( τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ p ( r ( σ )) G ( y ) d λ s S ( σ ) dγ u p ( s ) dµ (0) G ( u ) Let f 1 b e a function on G , defined b y the formula f 1 ( y ) = Z T (0) Z T χ B ( y q ( τ )) χ C ( τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) . F rom Lemma 7.3 o f [4] w e know that a system of measures λ • on a group oid G is left inv aria n t if and only if for an y x ∈ G and eve ry non-negative Borel f unction f on G , (5) Z f ( xy ) dλ d ( x ) ( y ) = Z f ( y ) d λ r ( x ) ( y ) . This implies, using x = p ( σ ) − 1 and the ab ov e f 1 , that Z G f 1 ( p ( σ ) − 1 y ) dλ p ( r ( σ )) G = Z G f 1 ( y ) d λ p ( d ( σ )) G . 24 A. CENS OR A ND D. GRA NDINI Therefore, returning to our main calculation and noting that d ( p ( σ ) − 1 y ) = d ( y ), we hav e µ − 1 P ( E ) = Z G (0) Z S (0) Z S χ A ( σ − 1 ) Z G Z T (0) Z T χ B ( y q ( τ )) χ C ( τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ p ( d ( σ )) G ( y ) d λ s S ( σ ) dγ u p ( s ) dµ (0) G ( u ) Using the fa ct that γ • p is a disin tegratio n of µ (0) S with resp ect to µ (0) G , follow ed b y Lemma 2.4, w e get = Z S (0) Z S χ A ( σ − 1 ) Z G Z T (0) Z T χ B ( y q ( τ )) χ C ( τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ p ( d ( σ )) G ( y ) d λ s S ( σ ) dµ (0) S ( s ) = Z S χ A ( σ − 1 ) Z G Z T (0) Z T χ B ( y q ( τ )) χ C ( τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ p ( d ( σ )) G ( y ) d µ S ( σ ) = Z S Z G Z T (0) Z T χ A ( σ − 1 ) χ B ( y q ( τ )) χ C ( τ − 1 ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ p ( d ( σ )) G ( y ) d µ S ( σ ) The measure µ (0) S is quasi-in v ariant. Therefore, form ula (2 ) of Remark 2.7 p ermits us to replace σ − 1 b y σ at the price of inserting ∆ − 1 S ( σ ): = Z S Z G Z T (0) Z T χ A ( σ ) χ B ( y q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ p ( r ( σ )) G ( y ) d µ S ( σ ) Re-expanding dµ S and then using Lemma 5.9 again, fo llow ed by Lemma 2.4, w e hav e = Z G (0) Z S (0) Z S Z G Z T (0) Z T χ A ( σ ) χ B ( y q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ p ( r ( σ )) G ( y ) d λ s S ( σ ) dγ u p ( s ) dµ (0) G ( u ) = Z G (0) Z G Z S (0) Z S Z T (0) Z T χ A ( σ ) χ B ( y q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) = Z G Z S (0) Z S Z T (0) Z T χ A ( σ ) χ B ( y q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dµ G ( y ) W e now use the quasi-inv aria nce of µ (0) G and form ula (2) of Remark 2.7 to write = Z G Z S (0) Z S Z T (0) Z T χ A ( σ ) χ B ( y − 1 q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( y ) d λ t T ( τ ) dγ r ( y ) q ( t ) dλ s S ( σ ) dγ d ( y ) p ( s ) dµ G ( y ) Next, w e apply the c haracterization (4) preceding Lemma 5 .9 ab o ve to the comp ositions S r S λ • S / / S (0) p γ • p / / G (0) and T r T λ • T / / T (0) q γ • q / / G (0) . W e obtain = Z G Z S Z T χ A ( σ ) χ B ( y − 1 q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( y ) d ( γ q ◦ λ T ) r ( y ) ( τ ) d ( γ p ◦ λ S ) d ( y ) ( σ ) dµ G ( y ) WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 25 W e can now use F ubini’s theorem, after whic h w e re- expand the comp ositions as well as µ G : = Z G Z T Z S χ A ( σ ) χ B ( y − 1 q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( y ) d ( γ p ◦ λ S ) d ( y ) ( σ ) d ( γ q ◦ λ T ) r ( y ) ( τ ) dµ G ( y ) = Z G (0) Z G Z T (0) Z T Z S (0) Z S χ A ( σ ) χ B ( y − 1 q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( y ) d λ s S ( σ ) dγ d ( y ) p ( s ) dλ t T ( τ ) dγ r ( y ) q ( t ) dλ u G ( y ) d µ (0) G ( u ) By Lemma 5.9 with T , t, τ and q in place of S, s, σ and p , w e get = Z G (0) Z T (0) Z T Z G Z S (0) Z S χ A ( σ ) χ B ( y − 1 q ( τ )) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( y ) d λ s S ( σ ) dγ d ( y ) p ( s ) dλ q ( r ( τ )) G ( y ) d λ t T ( τ ) dγ u q ( t ) dµ (0) G ( u ) = Z G (0) Z T (0) Z T Z G Z S (0) Z S χ A ( σ ) χ − 1 B ( q ( τ ) − 1 y ) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( y ) d λ s S ( σ ) dγ d ( y ) p ( s ) dλ q ( r ( τ )) G ( y ) d λ t T ( τ ) dγ u q ( t ) dµ (0) G ( u ) Let f 2 b e a function on G , defined b y the formula f 2 ( y ) = Z S (0) Z S χ A ( σ ) χ − 1 B ( y ) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ))∆ − 1 G ( y ) d λ s S ( σ ) dγ d ( y ) p ( s ) . Using x = q ( τ ) − 1 and f 2 in Equation (5) ab o v e, w e obtain that Z G f 2 ( q ( τ ) − 1 y ) dλ q ( r ( τ )) G = Z G f 2 ( y ) d λ q ( d ( τ )) G . Recall that w e tak e ∆ G to b e a group oid homomorphism (see Remark 2.7). Therefore, ∆ − 1 G ( q ( τ ))∆ − 1 G ( q ( τ ) − 1 y ) = ∆ − 1 G ( q ( τ ))∆ − 1 G ( q ( τ ) − 1 )∆ − 1 G ( y ) = ∆ − 1 G ( y ). Hence, noting also tha t d ( q ( τ ) − 1 y ) = d ( y ), the left hand side of the ab ov e equality give s precisely the last line of our main calculation. F rom the right ha nd side w e then get µ − 1 P ( E ) = Z G (0) Z T (0) Z T Z G Z S (0) Z S χ A ( σ ) χ B − 1 ( y ) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ))∆ − 1 G ( y ) dλ s S ( σ ) dγ d ( y ) p ( s ) dλ q ( d ( τ )) G ( y ) d λ t T ( τ ) dγ u q ( t ) dµ (0) G ( u ) F rom the fact that γ • q is a disin tegration o f µ (0) T with resp ect to µ (0) G , follow ed by Lemma 2.4 , w e get = Z T (0) Z T Z G Z S (0) Z S χ A ( σ ) χ B − 1 ( y ) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ))∆ − 1 G ( y ) dλ s S ( σ ) dγ d ( y ) p ( s ) dλ q ( d ( τ )) G ( y ) d λ t T ( τ ) dµ (0) T ( t ) = Z T Z G Z S (0) Z S χ A ( σ ) χ B − 1 ( y ) χ C ( τ − 1 )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ))∆ − 1 G ( y ) dλ s S ( σ ) dγ d ( y ) p ( s ) dλ q ( d ( τ )) G ( y ) d µ T ( τ ) 26 A. CENS OR A ND D. GRA NDINI Using the quasi-inv ar ia nce of µ (0) T and formula (2) of Remark 2 .7 gives = Z T Z G Z S (0) Z S χ A ( σ ) χ B − 1 ( y ) χ C ( τ )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ) − 1 )∆ − 1 G ( y )∆ − 1 T ( τ ) dλ s S ( σ ) dγ d ( y ) p ( s ) dλ q ( r ( τ )) G ( y ) d µ T ( τ ) Re-expanding dµ T w e get: = Z G (0) Z T (0) Z T Z G Z S (0) Z S χ A ( σ ) χ B − 1 ( y ) χ C ( τ )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ) − 1 )∆ − 1 G ( y )∆ − 1 T ( τ ) dλ s S ( σ ) dγ d ( y ) p ( s ) dλ q ( r ( τ )) G ( y ) d λ t T ( τ ) dγ u q ( t ) dµ (0) G ( u ) W e inv oke Lemma 5.9 once aga in, with T , t, τ and q in place of S, s, σ and p . W e obtain = Z G (0) Z G Z T (0) Z T Z S (0) Z S χ A ( σ ) χ B − 1 ( y ) χ C ( τ )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ) − 1 )∆ − 1 G ( y )∆ − 1 T ( τ ) dλ s S ( σ ) dγ d ( y ) p ( s ) dλ t T ( τ ) dγ r ( y ) q ( t ) dλ u G ( y ) d µ (0) G ( u ) By Lemma 2.4 t his equals = Z G Z T (0) Z T Z S (0) Z S χ A ( σ ) χ B − 1 ( y ) χ C ( τ )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ) − 1 )∆ − 1 G ( y )∆ − 1 T ( τ ) dλ s S ( σ ) dγ d ( y ) p ( s ) dλ t T ( τ ) dγ r ( y ) q ( t ) dµ G ( y ) W e once a gain now use the quasi-inv a riance of µ (0) G and formula (2) of Remark 2 .7 to write = Z G Z T (0) Z T Z S (0) Z S χ A ( σ ) χ B ( y ) χ C ( τ )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ) − 1 )∆ − 1 G ( y − 1 )∆ − 1 G ( y )∆ − 1 T ( τ ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ t T ( τ ) dγ d ( y ) q ( t ) dµ G ( y ) Returning to χ E and using Lemma 2.4, w e get = Z G (0) Z G Z T (0) Z T Z S (0) Z S χ E ( σ , y , τ )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ) − 1 )∆ − 1 T ( τ ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ u G ( y ) d µ (0) G ( u ) As w e argued earlier in this calculation, w e can c hange t he o rder of in tegration: = Z G (0) Z G Z S (0) Z S Z T (0) Z T χ E ( σ , y , τ )∆ − 1 S ( σ )∆ − 1 G ( q ( τ ) − 1 )∆ − 1 T ( τ ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) Finally , w e define Λ( σ, y , τ ) = ∆ − 1 S ( σ )∆ − 1 G ( q ( τ ) − 1 )∆ − 1 T ( τ ). W e get: µ − 1 P ( E ) = Z G (0) Z G Z S (0) Z S Z T (0) Z T χ E ( σ , y , τ )Λ ( σ , y , τ ) dλ t T ( τ ) dγ d ( y ) q ( t ) dλ s S ( σ ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) By Lemma 5.1 0 this equals Z P χ E ( σ , x, τ ) Λ ( σ , x, τ ) d µ P ( σ , x, τ ) , proving the claim fo r an y elemen tary op en set. In order to complete the pro of , we need to sho w that the claim holds WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 27 for an y Borel set E ⊆ P . F o r this, w e will in vok e Lemma 2.24 of [4], as in the pro of o f Prop osition 4.4. F or an y Borel subset E , we define µ ( E ) = µ − 1 P ( E ) and ν ( E ) = Z P χ E ( α )Λ( α ) dµ P ( α ) . As in Lemma 2.3, since µ (0) P is lo cally finite and λ • P is a con tinuous Haar system, the induced measure µ P is lo cally finite, hence so is the measure µ . Th us ν is lo cally finite as w ell, since µ ( E ) = ν ( E ) for an y elemen tary op en set E , and these sets constitute a basis B for the top ology of P by Remark 3 .2. Finally , µ and ν agree on finite in tersections of sets in B as these are themselv es elemen tary op en sets, so Lemma 2.24 o f [4] implies that µ ( E ) = ν ( E ) for all Bo rel sets. The pro of is complete.  Remark 5.12. In p articular, it fo l lows fr o m the ab ove c alculation that the mo dular function of µ P is given by ∆ P ( σ , x, τ ) = ∆ S ( σ )∆ T ( τ ) / ∆ G ( q ( τ )) . 6. The we ak pullback of Haar groupoids W e return to the we ak pullback diagram, whic h w e ha v e now completed: P π S           π T   = = = = = = = = λ • P , µ (0) P λ • S , µ (0) S S p   = = = = = = = = T q           λ • T , µ (0) T G λ • G , µ (0) G In order for ( P , λ • P , µ (0) P ) to indeed b e the w eak pullbac k in the category HG , it must b e a Haar group oid in t he sense of D efinition 2.8, and the maps π S : P → S a nd π T : P → T need to b e homomorphisms of Haar group oids in the sense o f Definition 2 .9. The first fact is an immediate corollary of Theorem 4 .2 and Prop osition 5.1 1. The second fact is prov ed b elo w. Corollary 6.1. The gr oup oid ( P , λ • P , µ (0) P ) is a Haar gr oup oid. Prop osition 6.2. The maps π S : P → S and π T : P → T ar e homomorph i s ms of Haar gr o up oids . Pr o of. By lemma 3.8, the maps π S and π T are contin uous group oid homomorphisms. It remains to sho w that they are measure class preserving with resp ect to the induced measures. W e prov e first that ( π S ) ∗ ( µ P ) ∼ µ S . Let Σ ⊆ S b e a Borel subset. Using the definition of µ P , w e ha v e ( π S ) ∗ ( µ P )(Σ) = µ P ( π − 1 S (Σ)) = Z P (0) λ ( s,g ,t ) P ( π − 1 S (Σ)) dµ (0) P ( s, g , t ) . Observ e that π − 1 S (Σ) = { ( σ, x, τ ) ∈ P | σ ∈ Σ } = (Σ × G × T ) ∩ P . Substituting λ ( s,g ,t ) P = λ s S × δ g × λ t T according to Definition 4.1, and noting that systems of measures are concen trated 28 A. CENS OR A ND D. GRA NDINI on fib ers, we g et: λ ( s,g ,t ) P ( π − 1 S (Σ)) = λ ( s,g ,t ) P  (Σ × G × T ) ∩ P ( s,g ,t )  = ( λ s S × δ g × λ t T )  (Σ × G × T ) ∩ ( S s × { g } × T t )  (b y Lemma 3 .7) = λ s S (Σ) · δ g ( { g } ) · λ t T ( T ) = λ s S (Σ) λ t T ( T ) Therefore, using L emma 5.10 and then rewriting η y b y Prop osition 5.3, w e ha v e ( π S ) ∗ ( µ P )(Σ) = Z G Z P (0) λ s S (Σ) λ t T ( T ) dη y ( s, g , t ) d µ G ( y ) = Z G Z Z Z S (0) × G × T (0) λ s S (Σ) λ t T ( T ) dγ r ( y ) p ( s ) dδ y ( g ) dγ d ( y ) q ( t ) dµ G ( y ) = Z G Z Z S (0) × T (0) λ s S (Σ) λ t T ( T ) dγ r ( y ) p ( s ) dγ d ( y ) q ( t ) dµ G ( y ) W e use F ubini’s theorem, as we ll as Lemma 2 .4 , to o btain = Z G (0) Z G Z S (0) Z T (0) λ s S (Σ) λ t T ( T ) dγ d ( y ) q ( t ) dγ r ( y ) p ( s ) dλ u G ( y ) d µ (0) G ( u ) F urthermore, the fact that λ u G is supp orted on G u dictates that r ( y ) = u , hence w e get = Z G (0) Z G Z S (0) λ s S (Σ) Z T (0) λ t T ( T ) dγ d ( y ) q ( t ) dγ u p ( s ) dλ u G ( y ) d µ (0) G ( u ) = Z G (0) Z S (0) λ s S (Σ) Z G Z T (0) λ t T ( T ) dγ d ( y ) q ( t ) dλ u G ( y ) d γ u p ( s ) dµ (0) G ( u ) W e now define a function h 1 on G (0) b y h 1 ( u ) = Z G Z T (0) λ t T ( T ) dγ d ( y ) q ( t ) dλ u G ( y ) . Since λ t T ( T ) > 0 for an y t , the function h 1 ( u ) is strictly p ositiv e on G (0) . Returning to our main calculation, w e hav e: ( π S ) ∗ ( µ P )(Σ) = Z G (0) Z S (0) λ s S (Σ) h 1 ( u ) dγ u p ( s ) dµ (0) G ( u ) = Z G (0) Z S (0) λ s S (Σ) h 1 ( p ( s )) dγ u p ( s ) dµ (0) G ( u ) since γ u p is concen trated on p − 1 ( u ). Finally , γ • p is a disin tegratio n of µ (0) S with resp ect to µ (0) G , hence ( π S ) ∗ ( µ P )(Σ) = Z S (0) λ s S (Σ) h 1 ( p ( s )) dµ (0) S ( s ) On the other hand, µ S (Σ) = Z S (0) λ s S (Σ) dµ (0) S ( s ) . It follows that µ S (Σ) = 0 if and only if ( π S ) ∗ ( µ P )(Σ) = 0 . WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 29 W e turn to π T . Pro ving that ( π T ) ∗ ( µ P ) ∼ µ T will require a detour via the quasi-in v ariance of µ (0) G . Let Ω ⊆ T b e a Borel subset. T r acing the line o f argumen ts ab ov e, w e hav e ( π T ) ∗ ( µ P )(Ω) = µ P ( π − 1 T (Ω)) = Z P (0) λ ( s,g ,t ) P ( π − 1 T (Ω)) dµ (0) P ( s, g , t ) , where λ ( s,g ,t ) P ( π − 1 T (Ω)) = λ ( s,g ,t ) P  ( S × G × Ω) ∩ P ( s,g ,t )  = ( λ s S × δ g × λ t T )  ( S × G × Ω) ∩ ( S s × { g } × T t )  = λ s S ( S ) λ t T (Ω) Therefore, ( π T ) ∗ ( µ P )(Ω) = Z G Z P (0) λ s S ( S ) λ t T (Ω) dη y ( s, g , t ) d µ G ( y ) = Z G Z Z Z S (0) × G × T (0) λ s S ( S ) λ t T (Ω) dγ r ( y ) p ( s ) dδ y ( g ) dγ d ( y ) q ( t ) dµ G ( y ) = Z G Z T (0) Z S (0) λ s S ( S ) λ t T (Ω) dγ r ( y ) p ( s ) dγ d ( y ) q ( t ) dµ G ( y ) Using the quasi-inv ar ia nce of µ (0) G and formula (2) of Remark 2 .7, we g et = Z G Z T (0) Z S (0) λ s S ( S ) λ t T (Ω)∆ − 1 G ( y ) d γ d ( y ) p ( s ) dγ r ( y ) q ( t ) dµ G ( y ) Replacing γ r ( y ) q b y γ u q as b efore, and using Lemma 2.4 and F ubini’s theorem, w e get = Z G (0) Z G Z T (0) λ t T (Ω) Z S (0) λ s S ( S )∆ − 1 G ( y ) d γ d ( y ) p ( s ) dγ u q ( t ) dλ u G ( y ) d µ (0) G ( u ) = Z G (0) Z T (0) λ t T (Ω) Z G Z S (0) λ s S ( S )∆ − 1 G ( y ) d γ d ( y ) p ( s ) dλ u G ( y ) d γ u q ( t ) dµ (0) G ( u ) The function h 2 on G (0) defined by h 2 ( u ) = Z G Z S (0) λ s S ( S )∆ − 1 G ( y ) d γ d ( y ) p ( s ) dλ u G ( y ) is p ositive since λ s S ( S ) > 0 fo r an y s and the mo dular f unction ∆ G is p ositive . Returning to our main calculation, we hav e: ( π T ) ∗ ( µ P )(Ω) = Z G (0) Z T (0) λ t T (Ω) h 2 ( u ) dγ u q ( t ) dµ (0) G ( u ) = Z G (0) Z T (0) λ t T (Ω) h 2 ( q ( t )) dγ u q ( t ) dµ (0) G ( u ) since γ u q is concen tra t ed on q − 1 ( u ). Finally , γ • q is a disin tegratio n of µ (0) T with resp ect to µ (0) G , hence ( π T ) ∗ ( µ P )(Ω) = Z T (0) λ t T (Ω) h 2 ( q ( t )) dµ (0) T ( t ) 30 A. CENS OR A ND D. GRA NDINI On the other hand, µ T (Ω) = Z T (0) λ t T (Ω) dµ (0) T ( t ) . It follows that µ T (Ω) = 0 if a nd only if ( π T ) ∗ ( µ P )(Ω) = 0 . This completes the pro o f .  Recall our standing Assumption 5.1, b y which the maps p and q (restricted to the unit spaces) admit disin tegratio ns whic h a re lo cally b ounded. As w e sho w in t he fo llo wing prop o- sition, the map π S will automatically inherit this prop erty . Ho w ev er, in order to guar an tee that the map π T admits a disin tegration whic h is lo cally b ounded, w e will need another assumption. Assumption 6.3. We wil l a s sume that the mo dular function ∆ G is lo c al ly b ounde d on G , in the sense that for ev ery p o i n t x ∈ G ther e exist a ne i g hb orho o d U x and p ositive c onstants c x and C x such that c x < ∆ G ( y ) < C x for every y ∈ U x . Note that ∆ − 1 G is lo cally b ounded whenev er ∆ G is lo cally b ounded. Remark 6.4. If we assume that ∆ S and ∆ T ar e al s o lo c al ly b ounde d in the ab ove sense, then R emark 5.12 implies that ∆ P is lo c al ly b ounde d as w e l l. Prop osition 6.5. The m aps π S : P (0) → S (0) and π T : P (0) → T (0) admit di s inte gr ations which ar e lo c al ly b ounde d. Pr o of. W e start with the map π S . W e shall use Prop osition 6.8 from [4], whic h provide s a necessary and sufficien t condition f or admitting a disin tegra tion whic h is lo cally b o unded: for any compact set K ⊆ P (0) there mus t exist a constant C K suc h that for all Bo rel sets Σ ⊆ S (0) , µ (0) P ( K ∩ π − 1 S (Σ)) ≤ C K · µ (0) S (Σ) . Let K ⊆ P (0) b e compact. Consider three increasing sequences { A n } , { B n } and { C n } of op en subsets with compact closures in S , G and T respective ly , suc h that S = S ∞ n =1 A n , G = S ∞ n =1 B n , and T = S ∞ n =1 C n (suc h sequences exist in a n y lo cally compact second coun table space). The elemen tar y op en sets E n = ( A n × B n × C n ) ∩ P (0) determine an increasing o p en co ve r of P (0) and in particular of K . Since K is compact, K ⊆ E i for some i . Denoting K 1 = A i , K 2 = B i and K 3 = C i , w e hav e K ⊆ ( K 1 × K 2 × K 3 ) ∩ P (0) where K 1 ⊆ S , K 2 ⊆ G and K 3 ⊆ T ar e each compact. F or any Bor el set Σ ⊆ S (0) , µ (0) P ( K ∩ π − 1 S (Σ)) ≤ µ (0) P (( K 1 × K 2 × K 3 ) ∩ π − 1 S (Σ)) = µ (0) P ((( K 1 ∩ Σ) × K 2 × K 3 ) ∩ P (0) ) = Z K 2 γ r ( x ) p ( K 1 ∩ Σ) γ d ( x ) q ( K 3 ) dµ G ( x ) where the last eq ualit y follo ws from a calculation a s in the pro of of Prop osition 5.6. Ex- panding µ G w e get = Z G (0) Z K 2 γ r ( x ) p ( K 1 ∩ Σ) γ d ( x ) q ( K 3 ) dλ u G ( x ) dµ (0) G ( u ) ≤ Z G (0) Z K 2 γ r ( x ) p (Σ) γ d ( x ) q ( K 3 ) dλ u G ( x ) dµ (0) G ( u ) WEAK PULLBACKS OF TOPOLOGICAL GROUPOIDS 31 Next, w e note t ha t r ( x ) = u since λ u G is supp orted on r − 1 ( u ), and t hen rewrite γ u p (Σ): = Z G (0) Z K 2 γ u p (Σ) γ d ( x ) q ( K 3 ) dλ u G ( x ) dµ (0) G ( u ) = Z G (0) Z K 2 Z S (0) χ Σ ( s ) γ d ( x ) q ( K 3 ) dγ u p ( s ) dλ u G ( x ) dµ (0) G ( u ) W e use F ubini’s Theorem a nd note that p ( s ) = u s ince γ u p is supp orted on p − 1 ( u ), after whic h w e can collapse the outer t w o in tegrals, since γ p is a disin t egr a tion: = Z G (0) Z S (0) Z K 2 χ Σ ( s ) γ d ( x ) q ( K 3 ) dλ p ( s ) G ( x ) dγ u p ( s ) dµ (0) G ( u ) = Z S (0) Z K 2 χ Σ ( s ) γ d ( x ) q ( K 3 ) dλ p ( s ) G ( x ) dµ (0) S ( s ) ≤ C · µ (0) S (Σ) , where C =  sup u γ u q ( K 3 )  ·  sup v λ v G ( K 2 )  . Bot h suprema exist since γ • q and λ • G are lo cally b ounded, hence b ounded on compact sets. W e turn to the map π T . The pro of will b e analogous, but will require the use o f the function ∆ − 1 G , whic h is lo cally b ounded by Assumption 6.3. Let Ω ⊆ T (0) . µ (0) P ( K ∩ π − 1 T (Ω)) ≤ µ (0) P (( K 1 × K 2 × K 3 ) ∩ π − 1 T (Ω)) = µ (0) P (( K 1 × K 2 × ( K 3 ∩ Ω)) ∩ P (0) ) = Z K 2 γ r ( x ) p ( K 1 ) γ d ( x ) q ( K 3 ∩ Ω) dµ G ( x ) = Z K − 1 2 γ d ( x ) p ( K 1 ) γ r ( x ) q ( K 3 ∩ Ω)∆ − 1 G ( x ) dµ G ( x ) Skipping interme diate calculations which mimic the π S case, w e get ≤ Z T (0) Z K − 1 2 χ Ω ( t ) γ d ( x ) p ( K 1 )∆ − 1 G ( x ) dλ q ( t ) G ( x ) dµ (0) T ( t ) ≤ D · µ (0) T (Ω) where D =  sup u γ u p ( K 1 )  · sup x ∈ K − 1 2 ∆ − 1 G ( x ) ! ·  sup v λ v G ( K − 1 2 )  . All suprema exist since γ • p and λ • G are b ounded on compact sets, and ∆ − 1 G is lo cally b ounded.  A cknowledgments W e thank John Baez, Christopher W alk er and most of all P aul Muhly for inspiring dis- cussions and useful remarks. Reference s [1] Claire Anan thara man-Delaro che and Jean Renault, Ame nable gr oup oids , Monogr aphies de L’Enseigne- men t Math´ ematique, volume 36, Genev a, 200 0. [2] John C. Baez, Alexander E. Hoffnung and Christopher D. W a lk er, Higher-dimensional algebr a VII: gr oup oidific ation , 2 009, prepr in t arXiv:0 908.430 5 v2 [math.Q A]. [3] ´ Etienne Blanchard, D´ eformations de C ∗ -alg ` ebr es de Hopf , Bull. So c. Math. F rance 124 (19 96), pp. 141–2 15. [4] Aviv C e ns or and Daniele Grandini, Bor el and c ontinuous systems of me asur es , 2010, preprint arXiv:100 4.3750 v1 [math.F A]. [5] P eter Hahn, Haar me asur e for me asure gr oup oids , T r ans. Amer. Math. So c. vol. 242 (1978 ), pp. 1–3 3. 32 A. CENS OR A ND D. GRA NDINI [6] George W. Mack ey , Er go dic the ory, gr oup the ory, and differ ential ge ometry , Pro c. Nat. Acad. Sci. U.S.A. 50 (1963), pp. 118 4–1191 [7] P aul S. Muhly , Co or dinates in op er ator algebr as , to app ear in CBMS lectur e no tes series . [8] Alan L. T. Paterson, Gr oup oids, inverse semigr oups, and their op er ator algebr as , Pr ogress in Mathe- matics, volume 170 , Birkhauser , B oston, 199 9. [9] Arlan B. Ramsay , Polish gr oup oids , in Descriptive set the ory and dynamic al systems , London Math. So c. L e cture Note Ser ies, volume 27 7, pp. 259– 271, Ca mbridge University Press, C a m br idge, 2000 . [10] Arlan B. Ramsay , Virtual gr oups and gr ou p actions , Adv ances in Math. 6 (1971 ), pp. 2 53–322 . [11] Jean Renault, A gr oup oid appr o ach to C ∗ -algebr as , Lecture Notes in Mathematics, volume 7 93, Springer, Berlin, 198 0. [12] An thony K . Seda , On me asur es in fibr e sp ac es , Cahiers de T op ologie es t G´ eom´ etrie Diff erentielle Cat´ ego riques, vol.21 no.3 (19 80) pp. 24 7–276. A viv Censor, Schoo l of Ma thema tical Sciences, Tel-A viv U niversity, Tel-A viv 69978 , I s- rael E-mail addr ess : avi vc@post .tau.ac.il Daniele Grandini, Dep ar tment o f Ma thema tics, University of California a t Riverside, Riverside, CA 92521 , U .S.A. E-mail addr ess : dan iele@ma th.ucr.edu