Measurable Hall's theorem for actions of abelian groups

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups, if two equidistributed measurable sets are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, M\'ath\'e and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.


Introduction
In 1925 Tarski famously asked if the unit square and the disk of the same area are equidecomposable by isometries of the plane, i.e. if one can partition one of them into finitely many pieces, rearrange them by isometries and obtain the second one. This problem became known as the Tarski circle squaring problem.
The question whether two sets of the same measure can be partitioned into congruent pieces has a long history. At the beginning of the 19th century Wallace, Bolyai and Gerwien showed that any two polygons in the plane of the same area are congruent by dissections (see [31,Theorem 3.2]) and Tarski [29] ( [31,Theorem 3.9]) showed that such polygons are equidecomposable using pieces which are polygons themselves. Hilbert's 3rd problem asked if any two polyhedra of the same volume are equidecomposable using polyhedral pieces. The latter was solved by Dehn (see [1]). Banach and Tarski showed that in dimension at least 3, any two bounded sets in R n with nonempty interior, are equidecomposable, which leads to the famous Banach-Tarski paradox on doubling the ball. Back in dimension 2, the situation is somewhat different, as any two measurable subsets equidecomposable by isometries must have the same measure (see [31]) and this was one of the motivation for the Tarski circle squaring problem. Using isometries was also essential as von Neumann [32] showed that the answer is positive if one allows arbitrary area-preserving transformations. The crucial feature that makes the isometries of the plane special is the fact that the group of isometries of R 2 is amenable. Amenability was, in fact, introduced by von Neumann in the search of a combinatorial explanation of the Banach-Tarski paradox.
The first partial result on the Tarski circle squaring was a negative result of Dubins, Hirsch and Karush [5] who showed that pieces of such decompositions cannot have smooth boundary (which means that this cannot be performed using scissors). However, the full positive answer was given by Laczkovich in his deep paper [16]. In fact, in [19] Laczkovich proved a stronger result saying that whenever A and B are two bounded measurable subsets of R n of positive measure such that the upper box dimension of the boundaries of A and B is less than n, then A and B are equidecomposable. The assumption on the boundary is essential since Laczkovich [17] (see also [18]) found examples of two measurable sets of the same area which are not equidecomposable even though their boundaries have even the same Hausdorff dimension. The proof of Laczkovich, however, did not provide any regularity conditions on the pieces used in the decompositions. Given the assumption that A and B have the same measure, it was natural to ask if the pieces can be chosen to be measurable. Moreover, the proof of Laczkovich used the axiom of choice.
A major breakthrough was achieved recently by Grabowski, Máthé and Pikhurko [10] who showed that the pieces in Laczkovich's theorem can be chosen to be measurable: whenever A and B are two bounded subsets of R n of positive measure such that the upper box dimension of the boundaries of A and B are less than n, then A and B are equidecomposable using measurable pieces. Another breakthrough came even more recently when Marks and Unger [25] showed that for Borel sets, the pieces in the decomposition can be even chosen to be Borel, and their proof did not use the axiom of choice.
The goal of the present paper is to give a combinatorial explanation of these phenomena. There are some limitations on how far this can go because already in Laczkovich's theorem there is a restriction on the boundary of the sets A and B. Therefore, we are going to work in the measure-theoretic context and provide sufficient and necessary conditions for two sets to be equidecomposable almost everywhere. Recently, there has been a lot of effort to develop methods of the measurable and Borel combinatorics (see for instance the upcoming monograph by Marks and Kechris [22]) and we would like to work within this framework.
The classical Hall marriage theorem provides sufficient and necessary conditions for a bipartite graph to have a perfect matching. Matchings are closely connected with the existence of equidecompositions and both have been studied in this context. In 1996 Miller [27,Problem 15.10] asked whether there exists a Borel version of the Hall theorem. The question posed in such generality has a negative answer as there are examples of Borel graphs which admit perfect matchings but do not admit measurable perfect matchings. One example is provided already by the Banach-Tarski paradox (see [22]) and Laczkovich [15] constructed a closed graph which admits a perfect matching but does not have a measurable one. In the Baire category setting, Marks and Unger [23] proved that if a bipartite Borel graph satisfies a stronger version of Hall's condition with an additional ε > 0, i.e. if the set of neighbours of a finite set F is bounded from below by (1 + ε)|F |, then the graph admits a perfect matching with the Baire property (see also [24] and [4] for related results on matchings in this context). On the other hand, in all the results of Laczkovich [19], Grabowski, Máthé and Pikhurko [10] and Marks and Unger [25] on the circle squaring, a crucial role is played by the strong discrepancy estimates, with an ε > 0 such that the discrepancies of both sets are bounded by C 1 n 1+ε (for definitions see Section 2). Recall that given a finitely generated group Γ generated by a symmetric set S and acting freely on a space X, the Schreier graph of the action is the graph connecting two points x and y if γ·x = y for one of the generators γ ∈ S.
is a free pmp action of a finitely generated group on a space X. Write G for the Schreier graph of the action. A pair of sets A, B satisfies the Hall condition (µ-a.e.) with respect to Γ (given a set of generators) if for every (µ-a.e.) x ∈ X and for every finite subset F of Γ · x we have This definition clearly depends on the choice of generators, and we say that A, B satisfy the Hall condition (µ-a.e.) if the above holds for some choice of generators. For the case with a fixed set of generators (which will be more natural for us), we say that the action Γ (X, µ) satisfies k-Hall condition (µ-a.e.) if for every (µ-a.e., resp.) x ∈ X for every finite subset F of Γ · x we have Note that A, B satisfy the Hall condition if and only if A, B satisfy the k-Hall condition for some k > 0. We will work under the assumption that both sets A, B satisfy certain form of equidistribution on the orbits, namely that they are Γ-uniform (for definition see Section 2).
Our main result is the following.
Theorem 2. Let Γ be a finitely generated abelian group and Γ (X, µ) be a free pmp action. Suppose A, B ⊆ X are two measurable Γ-uniform sets. The following are equivalent: (1) the pair A, B satisfies the Hall condition with respect to Γ µ-a.e., (2) A and B are Γ-equidecomposable µ-a.e. using µ-measurable sets, (3) A and B are Γ-equidecomposable µ-a.e.
As a consequence, it gives the following.
Corollary 3. Suppose Γ is a finitely generated abelian group and Γ (X, µ) is a free pmp Borel action on a standard Borel probability space. Let A, B ⊆ X be measurable Γ-uniform sets. If A and B are Γ-equidecomposable, then A and B are Γ-equidecomposable using measurable pieces.
This generalizes the recent measurable circle squaring result [10] as Laczkovich [16] constructs an action of Z d satisfying the conditions above, for a suitably chosen d (big enough, depending on the box dimensions of the boundaries).
In fact, in 1991 Gardner [8,Conjecture 6] conjectured that whenever two bounded measurable sets in an Euclidean space are equidecomposable via an action of an amenable group of isometries, then they are equidecomposable using measurable sets. The above corollary confirms this conjecture in case of an abelian group Γ and Γ-uniform sets.
The main new idea in this paper is an application of measurable medial means. These have previously used in descriptive set theory only in the work of Jackson, Kechris and Louveau [11] on amenable equivalence relations but that context was combinatorially different. 1 They are used together with a recent result of Conley, Jackson, Kerr, Marks, Seward and Tucker-Drob [3] on tilings of amenable group actions in averaging sequences of measurable matchings. This allows us to avoid using Laczkovich's discrepancy estimates that play a crucial role in both proofs of the measurable and Borel circle squaring. We also employ the idea of Marks and Unger in constructing bounded measurable flows. More precisely, following Marks and Unger we construct bounded integer-valued measurable flows from bounded real-valued measurable flows. However, instead of using Timár's result [30] for specific graphs induced by actions of Z d , we give a self-contained simple proof of the latter result, which works in the measurable setting for the natural Cayley graph of Z d . This is the only part of the paper which deals with abelian groups and we hope it could be generalized to a more general setting. On the other hand, the measurable averaging operators that we employ, cannot be made Borel and for this reason the results of this paper apply to the measurable setting and generalize only the results of [10].
While this paper deals with abelian groups (the crucial and only place which works under these assumptions is Section 6), a positive answer to the following question would confirm Gardner's conjecture [

Discrepancy estimates
Both proofs of Grabowski, Máthé and Pikhurko and of Marks and Unger use a technique that appears in Laczkovich's paper [16] and is based on discrepancy estimates. Laczkovich constructs an action of a group of the form Z d for d depending on the upper box dimension of the boundaries of the sets A and B such that both sets are very well equidistributed on orbits on this action. To be more precise, given an action Z d (X, µ) and a measurable set A ⊆ X, the discrepancy of A with respect to a finite subset F of an orbit of the action is defined as It is meaningful to compute the discrepancy with respect to finite cubes, i.e. subsets of orbits which are of the form [0, n] d · x, where x ∈ X and [0, n] d ⊆ Z d is the d-dimensional cube with side {0, . . . , n}. The cube [0, n] d has boundary, whose relative size with respect to the size of the cube is bounded by c · 1 n for a constant c.
A crucial estimation that appears in Laczkovich's paper is that the action of Z d is such that for both sets A and B the discrepancy is estimated as for some ε > 0 and some c > 0, which means that the discrepancies of both sets on cubes decay noticeably faster than the sizes of the boundaries of these cubes.
A slightly more natural condition on the equidistrubition of a set A would be to remove the ε from ( * ) and require that there exists a constant c such that for every n the discrepancy In this paper, we work with even weaker assumption on equidistribution, given by the following definition.

Definition 5. Given a Borel free pmp action
Note that this definition does not depend (up to changing the constant c) on the way the group is written as Z d × ∆ and a choice of generators for the group.

Measurable averaging operators
In this paper we use special kinds of measurable averaging operators. These can be constructed in different ways.
For the first construction, recall the definition of a medial mean.

Definition 6. A medial mean is a linear functional
Medial means were studied already by Banach who showed their existence (the so-called Banach limits). In this paper we use a special kind of medial means m which is additionally measurable on [0, 1] N and we use it to take a measurable average of a sequence of functions f n : (X, µ) → [0, 1] for a space (X, µ) with a Borel probability measure µ. It takes a bit more effort to construct medial means that are measurable but this can be done in a couyple of ways.
Recall that Mokobodzki showed that under the assumption of the Continuum Hypothesis there exists a medial mean which is universally measurable as a function on [0, 1] N . For a proof the reader can consult the textbook of Fremlin [6, Theorem 538S] or the article [26]. However, a careful analysis of Mokobodzki's proof shows that for a single Borel probability measure µ, the existence of a µ-measurable medial mean does not require the Continuum Hypothesis. Nevertheless, some settheoretical assumptions are still used even for a single µ, such as the Hahn-Banach theorem.
Another construction of measurable averaging a sequence of measurable functions f n : X → [0, 1] (perhaps more familiar to the general mathematical audience than the Mokobodzki construction) can be done using the Banach-Saks theorem in L 2 (X, µ) by using weak * -compactness of the unit ball (cf. [21]). This also needs some weak form of the Axiom of Choice such as the Tichonov theorem.

Set-theoretical assumptions
In view of the foundational questions and the role of the Axiom of Choice in equidecompositions (e.g. recall that the Hahn-Banach theorem implies the Banach-Tarski paradox [28]), we argue below that for the measurable equidecompositions on co-null sets, we can remove any set theoretic assumptions beyond ZF and the Axiom of Dependent Choice (DC) that are needed to obtain a measurable medial mean.
Recall that Borel sets can be coded using a Π 1 1 set (of Borel codes) BC ⊆ 2 N in a ∆ 1 1 way, i.e. there exists a subset C ⊆ BC × X such that the family {C x : x ∈ BC} consists of all Borel subsets of X and the set C can be defined using both Σ 1 1 and Π 1 1 definitions. For details the reader can consult the textbook of Jech [12,Chapter 25].
Given a Borel probability measure µ on X and a subset P ⊆ X × Y , we write The proposition below implies that if two Borel sets are coded using a real r, then we can argue about their measurable equidecomposition a.e. in L[r] (where AC and CH hold).
Borel space X with a Borel probability measure µ, two Borel subsets A, B ⊆ X and Γ (X, µ) is a Borel pmp action of a countable group Γ. The statement that the sets A and B are Γ-equidecomposable µ-a.e. using µ-measurable pieces is absolute between V and W .
Proof. Suppose that in W or V the sets A and B are Γ-equidecomposable µ-a.e. Then there exist disjoint Borel subsets A 1 , . . . A n of A and disjoint Borel subsets This statement can be written as and thus is it Σ 1 2 . By Shoenfield's absoluteness theorem [12,Theorem 25.20], it is absolute between V and W .

Measurable flows in actions of amenable groups
Given a standard Borel space X, a Borel graph G on X and f : Let Γ be a finitely generated amenable group. Let γ 1 , . . . , γ d be a finite symmetric set of generators of Γ. Let X be a standard Borel space and let µ be a Borel probability measure on X. Let Γ (X, µ) be a free pmp action. Recall that by the Schreier graph of the action we mean the graph In the following lemma we assume that there exists a universally measurable medial mean m, which, by the remarks in the previous section, we can assume throughout this paper.
In order to make it a bit more general, let us define the Hall condition for functions: a function f : X → Z satisfies the k-Hall condition if for every finite set F contained in an orbit of Γ X we have that Note that a pair of sets A, B satisfies the k-Hall condition if and only if f = χ A −χ B satisfies the k-Hall condition.
Proposition 9. Let Γ be a finitely generated amenable group and Γ (X, µ) be a Borel free pmp action.
where d is the number of generators of Γ.
Proof. First, we are going to assume that |f | ≤ 1, i.e. that f = χ A − χ B for two measurable subsets A, B ⊆ X. Indeed, replace X with X × l and take the projection π : X × l → X. Then we can find two subsets A, B ⊆ X × l such that We can also induce the graph structure on X × l by taking as edges all the pairs ((x, i), (y, j)) such that (x, y) forms an edge in X as well as all pairs ((x, i), (x, j)) for i = j. Then A and B satisfy the k-Hall condition in X × l for the above graph. Let K = {γ ∈ Γ : d(e, γ) ≤ k}. Fix δ > 0. Use the Conley-Jackson-Kerr-Marks-Seward-Tucker-Drob tiling theorem [3, Theorem 3.6] for K and δ to get a µ-conull Γ-invariant Borel set X ′ ⊆ X, a collection {C i : 1 ≤ i ≤ m} of Borel subsets of X ′ , and a collection {F i : are finite subsets of Γ and x, y ∈ X then F ′ (K)x = F ′′ (K)y. If F ⊆ X is a finite subset of a single orbit, then we let F (K) = F ′ (K)x where F ′ ⊆ Γ and x ∈ X satisfy F = F ′ x. This definition does not depend on the choice of representation F = F ′ x by the previous remark. Note that if F ⊆ X is (K, δ)-invariant then Then H is a locally finite Borel graph satisfying Hall's condition as A, B satisfy the k-Hall condition. By the Hall theorem, there exists a Borel injection x 2 ), . . . , (x j−1 , x j )} be the shortest lexicographically smallest path in the graph G connecting Define φ : G → R by the formula φ(x, γx) = |{p ∈ P : (x, γx) ∈ p}| − |{p ∈ P : (γx, x) ∈ p}|.
Note that φ is Borel (by definition). Also, |φ| is bounded by d k (the number of paths of length not greater than k passing through a given edge in the graph G).
Note that for every x ∈ X ′′ we have Then {C 1,i , C 2,i , . . . , C ni,i } is a partition of C i into Borel sets.
Observe that for every F ∈ F we have It follows that Now, for every n pick δ n > 0 so that 1 − 3δ n |K| > 1 − 1 2 n . Denote h n = h, φ n = φ and X n = X ′′ where h, φ and X ′′ are constructed above for this particular δ n . Let ∞ n=m X n . Then µ(Y ) = 1. We can assume that Y is Γ-invariant (by taking its subset if needed). Denote by G the Schreier graph of where m denotes the medial mean. Then for x ∈ Y we have Moreover, |φ| is bounded by d k , which is a common bound for the flows φ n . For measurability of φ, write µ ′ = φ * (µ×µ) for the pushforward to [−d k , d k ] N of the measure µ×µ on the graph G and note that since m is µ ′ -measurable, it follows that φ is µ-measurable.

Flows in Z d
In this section we prove a couple of combinatorial lemmas which lead to a finitary procedure of changing a real-valued flow on a cube in Z d to an integer-valued flow on a cube in Z d . This gives an alternative proof of [25,Lemma 5.4] in the measurable setting. Also, this is the only part of the paper which deals with the groups Z d as opposed to arbitrary amenable groups. Let So, E(A) is the set of positively oriented edges whose both endpoints are in A, E + (A) is the set of positively oriented edges whose at least one endpoint is in A, and N (A) is the neighbourhood of A (in the sup-norm).
Definition 11. We say that a subset C of Z d is a cube if C is of the form That is, x1,x2,x3,x4 s is a flow sending s units through the path s is an f -flow such that |ϕ − ψ| < 1 and ψ(x 1 , x 4 ) is an integer. We will now prove a couple of lemmas stating that one can modify a flow so that it becomes integer-valued on certain sets of edges.
for all x ∈ i<j C i . So, let ϕ 0 = ϕ. Given ϕ j we define ϕ j+1 in the following way. For every x ∈ C j let x = x,y,z,t s where y = x + e ℓ , z = y + e d , t = z − e ℓ = x + e d and Also, the sets supp( x ) are pairwise disjoint for x ∈ C j , and therefore, by definition of ϕ j+1 we have for It is also clear that supp( x ) ⊆ E(C), so Therefore ϕ j+1 satisfies all required properties.

For (ii) observe that
By Lemma 13, ϕ j agrees with ϕ j−1 on {(x, x+e d ) : x ∈ D j−1 }, thus ϕ j (x, x+e d ) ∈ Z for x ∈ D j−1 . Moreover, ϕ j (x, x + e d ) ∈ Z for x ∈ D j \ D j−1 again by Lemma 13. Also (iii) is immediate by Lemma 13. Therefore ϕ j satisfies the required properties.
We define ψ = ϕ d−1 . By construction, ψ satisfies the first two conditions. For the third condition note that Lemma 15. Let C be a cube. Let C be a collection of cubes such that: • Let f : Z d → Z. Let ϕ : G → R be a bounded f -flow. Then there exists an f -flow ψ : G → R such that: • ψ(e) is integer for every edge e ∈ E, • |ϕ − ψ| < 6d.
Proof. Without loss of generality we may assume that be the set of vertical edges from E having their starting point in H k and be the set of edges from E having both endpoints in H k .
Thus ϕ 2k+1 is as required. Now suppose that ϕ 2k+1 is defined. We construct ϕ 2k+2 (cf. Fig. 2). Let D = {x : (x, x + e d ) ∈ E 2k+2 }. Note that every x ∈ D is either an element of C \ {N (C ′ ) : C ′ ∈ C} or lies on the upper face of some cube N (C ′ ) for C ′ ∈ C. We also note that if C ′ ∈ C then the upper face of N (C ′ ) is either contained in D or disjoint from D. So, let C 1 , C 2 , . . . C n be all elements of C such that the upper faces D 1 , D 2 , . . . , D n of N (C 1 ), N (C 2 ), . . . , N (C n ) are subsets of D. Let Since f (x) ∈ Z and it follows that ϕ 2k+1 (x, x + e d ) ∈ Z.
Next we deal with the case x ∈ D j for some j ≤ n. Each D j , j ≤ n is dealt with separately. For every j ≤ n we obtain an f -flow ϕ ′ j by applying Lemma 14 for ϕ 2k+1 , f and the cube Indeed, observe that Since f (x) ∈ Z for every x and, by the properties of ϕ ′ j , we have that ϕ ′ j (x, y) ∈ Z for all (x, y) = (x ′ , x ′ + e d ) with x ∈ N (C j ) and y / ∈ N (C j ), it follows that ϕ ′ j (x ′ , x ′ + e d ) ∈ Z as well. We define ϕ 2k+2 by the formula ϕ 2k+2 is well-defined because E(D ′ j ) are pairwise disjoint. By definition, it is integer-valued on i≤2k+2 E i , and the conditions on supp(ϕ − ϕ 2k+2 ) are clearly satisfied. Thus ϕ 2k+2 is as required.
We put ψ = ϕ 2k d . It remains to check that |ϕ − ψ| < 6d. This follows from the fact that the value on every edge was modified at most three times by at most 2d.

Measurable bounded Z-flows a.e.
In this section we show how to turn a measurable bounded real-valued flow into a measurable bounded integer-valued flow on a set of measure 1. We only use Lemma 15 proved in the previous section and the Gao-Jackson tiling theorem for actions of Z d .
Suppose Z d (X, µ) is a free pmp action. We follow the notation from the previous section in the context of the action. Definition 17. Let C ⊆ [X] <∞ be a collection of cubes. We say that it is nested if for every distinct C, C ′ ∈ C:

Definition 16. We say that a finite subset of X is a cube if it is of the form
Definition 18. Given a cube of the form by its interior we mean the cube and its boundary is is a free pmp action. Then there is a sequence of familes F n of cubes such that each F n consists of disjoint cubes, F n is nested and covers X up to a set of measure zero.
Proof. If S and T are families of sets, define Note that (S ⊓ T ) = ( S) ∩ ( T ). Also note that if S and T are families of cubes then S ⊓ T is a family of cubes as well. We also write int S = {int C : C ∈ S} and int k for the k-th iterate of int.
Use the Gao-Jackson theorem [7] to obtain a sequence of partitions S 1 , S 2 , . . . of X so that S n consists of cubes with sides n 3 or n 3 + 1. Define S 1 n = int S n and S k n = S k−1 n ⊓ int k S n+k for k > 1. Note that each S k n consists of pairwise disjoint cubes.
Define F n = lim inf m S m n = {C : ∃m 0 ∀m ≥ m 0 C ∈ S m n }. Note that if C ∈ F n then there exist unique cubes C n ∈ S n , C n+1 ∈ S n+1 , . . . such that C = k≥0 int k+1 C n+k . Also note that F n = ∞ k=0 int k+1 S n+k . We claim that F = n F n is nested and covers a set of measure 1. For nestedness, consider cubes C, C ′ ∈ F . Then C ∈ F n , C ′ ∈ F m for some n, m. We may assume that m ≥ n. Write C = k≥0 int k+1 C n+k and C ′ = This shows that F is nested. We will prove now that µ( F ) = 1.
For a cube C let x C to be the point x ∈ X such that C = d i=1 [0, n i ] · x C . For a positive integer n write X n = {x C : C ∈ S n }. Note that for any 0 ≤ k < n This implies that Hence µ ( F ) = 1.
Marks and Unger [25,Lemma 5.4] showed that for every d ≥ 2, any Borel, bounded real-valued flow on the Schreier graph of a free Borel action of Z d can be modified to a bounded Borel integer-valued flow. Below we provide a short proof for the case d = 1 and additionally an independent proof (based on Lemma 15) for d ≥ 2 in the case of a pmp action where we consider flows defined a.e. Proposition 20. Suppose Z d (X, µ) is a free pmp action and G is its Schreier graph. Let f : X → Z be a bounded measurable function. For every measurable f -flow ϕ : G → R, there exists a measurable bounded ψ : G → Z such that: • ψ is an f -flow µ-a.e., • |ψ| ≤ |ϕ| + 12d.
Proof. First we deal with the case d = 1. In that case for every e ∈ G we simply put ψ(e) = ⌊ϕ(e)⌋. Note that since G is a graph of degree 2, for every x ∈ X, the fractional parts of the two edges which contain x are equal because f is integervalued. Thus, ψ is also an f -flow. Now suppose d ≥ 2. By Lemma 19, there exists an invariant subset X ′ ⊆ X of measure 1 and a sequence of families F n of cubes such that n∈N F n is nested, each F n consists of disjoint cubes, n∈N F n covers X ′ . By induction on n we construct measurable f -flows ϕ n such that ϕ 0 = ϕ and • ϕ m = ϕ n+1 for every m > n on every E + (C) for C ∈ F n , • |ϕ n | ≤ |ϕ| + 12d. Given the flow ϕ n we apply Lemma 15 on each cube C ∈ F n to obtain the flow ϕ n+1 . The bound on ϕ n follows from the fact that the value of the flow on each edge is changed at most twice by at most 6d along this construction.
The sequence ϕ n converges pointwise on the edges of X ′ to a measurable f -flow ϕ ∞ , which is integer-valued on all edges in X ′ except possibly for the edges in bd C for cubes C ∈ n F n . However, the family {bd C : C ∈ n F n } consists of pairwise disjoint finite sets. By the integral flow theorem for finite graphs, we can further correct ϕ ∞ on each of these finite subgraphs without changing the bound |ϕ| + 12d to obtain a measurable integer-valued f -flow ψ, which is equal to ϕ ∞ on all edges from G \ {E(bd C) : C ∈ n∈N F n }.

Hall's theorem
In this section we prove Theorem 2. The proof of (1)⇒(2) is based on an idea of Marks and Unger [25].
(3)⇒(1) is true for every finitely generated group Γ. In general, if A and B are Γ-equidecomposable, and the group elements used in the decomposition are γ 1 , . . . , γ n , then A and B satisfy the k-Hall condition for k greater than the word lengths of the group elements γ 1 , . . . , γ n . If X ′ ⊆ X is a set of measure 1 such that A ∩ X ′ and B ∩ X ′ are Γ-equidecomposable, then A ∩ X ′ and B ∩ X ′ satisfy the k-Hall condition.
(1)⇒(2). Without loss of generality assume that the k-Hall condition and Γuniformity is satisfied everywhere Let Γ = Z d × ∆ where ∆ is a finite group and d ≥ 0.
If d = 0, then the group Γ is finite and the action has finite orbits (the discrepancy condition trivializes and we do not need to use it). On each orbit the Hall condition is satisfied, so on each orbit there exists a bijection between A and B on that orbit. Thus, the sets A and B are ∆-equidecomposable using a Borel choice of bijections on each orbit separately.
Thus, we can assume for the rest of the proof that d ≥ 1. Since ∆ is finite, we can quotient by its action and get a standard Borel space X ′ = X/∆ with the probability measure induced by the quotient map π : X → X ′ . We then have a free pmp action of Z d X ′ . Consider the function f : Note that f is bounded by |∆|. Using Proposition 9 and Proposition 20 we get an invariant subset Y ′ ⊆ X ′ of measure 1 and an integer-valued measurable f -flow ψ on the edges of the Schreier graph G of Z d Y ′ on Y ′ such that |ψ| ≤ |∆| d k + 12d. Again, without loss of generality, we can assume Y ′ = X ′ by replacing X with Y = π −1 (Y ′ ), if needed.
Note that there exists a constant r, depending only on d such that for every tiling of Z d with cubes with sides n or n + 1, every cube is adjacent to at most r many other cubes in the tiling.
Note that Γ-uniformity implies that for every set D such that D = D ′ × ∆ where D ′ is a cube with sides n or n + 1 we have |A ∩ D|, |B ∩ D| ≥ cn d |∆|. Let n be such that cn d |∆| ≥ r(n + 1) d−1 (|∆| d k + 12d).
Using the Gao-Jackson theorem [7], find a Borel tiling T ′ of X ′ with cubes of sides n or n + 1. Pulling back the tiling to X via π, we get a Borel tiling T of X with cubes of the form D = (D ′ × ∆) · x where D ′ has sides of length n or n + 1. Note that for every tile D in T we have ( * ) |A ∩ D|, |B ∩ D| ≥ r(n + 1) d−1 (|∆| d k + 12d).
Let H be the graph on T where two cubes are connected with an edge if they are adjacent and similarly let H ′ be the graph on T ′ with two cubes connected with an edge if they are adjacent. We have two functions F ′ : T ′ → Z defined as F ′ (C) = x ′ ∈C f (x ′ ) and F : T → Z defined as F (C) = |A ∩ C| − |B ∩ C|.
Note that each vertex in H ′ has degree at most r and the same is true in H. Thus, by ( * ), for each C ∈ T and D ∈ T which are connected with an edge in H, we can find pairwise disjoint sets A(C, D), B(C, D) ⊆ C of size at least (n + 1) d−1 (|∆| d k + 12d) such that A(C, D) ⊆ A ∩ C, B(C, D) ⊆ B ∩ C. Now, the function which witnesses the equidecomposition is defined in two steps. First, we can find a map g : dom(g) → B such that dom(g) ⊂ (C,D)∈H A(C, D), for any two neighbouring cubes C, D ∈ T satisfying Ψ(C, D) > 0 we have | dom(g)∩ A(C, D)| = Ψ(C, D) and the points in dom(g) ∩ A(C, D) are mapped injectively to B(C, D). Note that for any cube C ∈ T the set C ∩ (A \ dom(g)) contains as many points as the set C ∩ (B \ im(g)). Hence, one can extend g to a funcion g ′ : A → B in such a way that its restriction to C ∩ (A \ dom(g)) is a bijection onto C ∩ (B \ im(g)) for any cube C ∈ T . Since ψ and hence Ψ ′ and Ψ are measurable, the function g ′ can be chosen to be measurable and it moves points by at most 2(|∆| + (n + 1) d ) in the Schreier graph distance. Thus, g ′ witnesses that A and B are equidecomposable using measurable pieces.

Measurable circle squaring
In this section we comment on how Corollary 3 follows from Theorem 2. We use an argument which appears in a preprint of Grabowski, Máthé and Pikhurko [9] and provide a short proof for completeness. Lemma 21. Suppose Γ (X, µ) is a free pmp action of a countable group Γ. If A, B ⊆ X are Γ-equidecomposable and X ′ ⊆ X is Γ-invariant, then A ∩ X ′ and B ∩ X ′ are also equidecomposable. If X ′ is additionally µ-measurable and A and B are Γ-equidecomposable using µ-measurable pieces, then A ∩ X ′ and B ∩ X ′ are Γ-equidecomposable using µ-measurable pieces.
Proof. The proof is the same in both cases. Let A 1 , . . . , A n and B 1 , . . . , B n be partitions of A and B such that γ i A i = B i for some γ i ∈ Γ. Put A ′ i = A i ∩ X ′ and B ′ i = B i ∩ X ′ . Then γ i A ′ i = B ′ i , so A ′ i and B ′ i witness that A ∩ X ′ and B ∩ X ′ are equidecomposable.
Lemma 22. Let µ be a probability measure on X and Γ X be a Borel pmp action of a countable group Γ. Suppose A, B ⊆ X are Γ-equidecomposable and there exists a measurable set Y ⊆ X of measure 1 such that A ∩ Y, B ∩ Y are equidecomposable using µ-measurable piecces. Then A, B are equidecomposable using µ-measurable pieces.