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Bounded Scale Measure and Property A

We introduce a generalization for bounded geometry that we call bounded scale measure. We show that bounded scale measure is a coarse invariant unlike bounded geometry. We then show equivalent definitions for spaces with bounded scale measure and show other properties that spaces with bounded scale measure satisfy. From there, we generalize property A from uniformly discrete metric spaces to large scale spaces with bounded geometry. Lastly, we construct a definition for property A for large scale spaces with bounded scale measure and show that this definition of property A is a coarse invariant.


Introduction
The main focus of this paper is to introduce a generalization of bounded geometry that we will call bounded scale measure. From there, we will introduce several useful equivalences of bounded scale measure and properties that spaces with bounded scale measure satisfy. Finally, we will define a generalization of property A for large scale structures with bounded geometry and for large scale spaces with bounded scale measure. We call this new generalization property A at scale U , and show that it is a coarse invariant. To begin, we will introduce the basic definitions needed to understand large scale structures. Definition 2. Let U , V be families of subsets of a set X. We say U is a refinement of V provided for every U ∈ U there is a V ∈ V so that U ⊆ V . In this same situation, we also say that V coarsens U . Refinement is denoted as U ≺ V.
It is sometimes needed to consider covers of X instead of collections of subsets of X. To distinguish families of subsets of X from covers of X, we call covers of X scales: Definition 3. Given a set X, we say U is a scale of X if U is a family of subsets of X that covers X. If U is a collection of subsets of X, we can make U into a cover via constructing U ′ = U ∪ {{x}} x∈X . This extension is often called the trivial extension of U .
The definition of large scale structures was given by Dydak and Hoffland in [2]. This equivalent interpretation of coarse structures allows coarse geometry to be approached in a more topological way.
Definition 4. Let X be a set. A large scale structure on X is a non-empty set of families LSS of subsets of X so that the following conditions are satisfied: 1. If we have families U , V of subsets of X with V ∈ LSS and each element U of U consisting of more than one point is contained in some V of V, then V ∈ LSS.
Elements U of LSS are called uniformly bounded families or uniformly bounded scales. If a large scale structure exists on a space X, we call the pair (X, LSS) a large scale space.
We note here closure under refinements implies the first condition above. The advantage of having a weaker first requirement is that a large scale structure as defined "disregards" one point sets. That is, one point sets do not "change" the large scale structure. Also, the first item in the definition gives us that the cover {{x}} x∈X is uniformly bounded for any large scale structure.
The following definition is also a nice example of large scale structures: Note that this definition also applies to metrics that are allowed to take on the value ∞. Such metrics are called ∞-metrics.
Remark 1.1. In [2], it was shown in Theorem 1.8 that a large scale structure induced by a metric is generated by the covers by n − balls for n ∈ N. This is to say that every uniformly bounded family of a large scale structure is a refinement of a cover of n − balls for some n. We will use this idea occasionally throughout the paper.
As is often the case with mathematical objects, we seek to compare one object to another. We do this through large scale continuous functions. Think of large scale continuous functions as analogous to continuous functions in topology. There are other types of large scale functions that are useful to know: We now wish to define something analogous to homeomorphisms in topology. However, there is a wrinkle. Homeomorphisms require functional inverses (i.e. the composition of f and its inverse in either order is the identity function on the appropriate set). However, in large scale geometry we can weaken the notion of requiring a functional inverse. This is because two distinct functions can appear similar should one "zoom out" far enough. That is, the outputs of the functions remain a uniformly bounded distance apart. Such functions are called close.
Definition 8. Let (X, LSS X ) and (Y, LSS Y ) be large scale spaces and let f, g ∶ X → Y . We say f and g are close provided there is a V ∈ LSS Y so that for any x ∈ X, f (x), g(x) ∈ V for some V ∈ V.
Notice how the following definition of our version of a homeomorphism differs only in that we require our functional compositions are close to the appropriate identity as opposed to being the identity functions themselves: Like in topology, we are interested in studying properties of large scale spaces that are preserved by coarse equivalences. We give a definition of such properties below: Definition 10. Let (X, LSS X ) be a large scale space. A property P is a coarse invariant if for any (Y, LSS Y ) that has property P and is coarsely equivalent to (X, LSS X ) we have that (X, LSS X ) also has property P .

Bounded Scale Measure
In order to motivate bounded scale measure, recall the definition of a large scale space having bounded geometry: Definition 11. Let (X, LSS) be a large scale space. We say (X, LSS) has bounded geometry if for every U ∈ LSS there exists an n ∈ N so that for all U ∈ U we have that U ≤ n.
One of the main issues with considering large scale spaces with bounded geometry is that bounded geometry is not a coarse invariant. Indeed, as in Example 1.2 (Z, LSS) and (R, LSS ′ ) are coarsely equivalent, but (Z, LSS) has bounded geometry while (R, LSS ′ ) does not. The idea for bounded scale measure came while trying to create a generalization of bounded geometry that is also a coarse invariant. The idea is to consider elements of a special scale, normally referred to as U , to be the "points" of the set, and use that to get a finite bounds on all other scales of X.
We will now give a definition of bounded scale measure with regards to maximal sets. From there, we will give two equivalent definitions that are useful for proving other properties that spaces with bounded scale measure satisfy. Finally, we will show that having bounded scale measure is a coarse invariant.
Definition 12. Given a scale U on X and given a subset A of X, a U -net in A is a maximal subset B of A with respect to the following property: no two elements of B belong to the same element of the scale U .
Definition 13. (X, LSS) has bounded scale measure at scale U if there exists a scale U such that for all scales V there exists a natural number u(V) such that for all V ∈ V any U -net of V has at most u(V) elements.
It turns out that this definition can be weakened somewhat. Having bounded scale measure at scale U currently requires that for any scale V and any element V of V has the property described for all U -nets of V . However, we will show that having bounded scale measure at scale U is equivalent to having a scale W so that for any scale V and element V of V, there exists a W-net of V that has the property described. Namely, the scale W is st(U , U ). Proof. First suppose there exists a scale U such that for all scales V there exists a natural number u(V) such that for all V ∈ V any U -net of V has at most u(V) elements. Let V be a scale of X and let V ∈ V. Take x ∈ V and then choose all points By construction this is a U -net in V and thus there exists k ∈ N with k ≤ u(V), as otherwise we have a U -net with more than u(V) elements. Thus taking our n(V) = u(V) and the collection {x i } 1≤i≤k satisfies our condition.
Conversely, suppose there exists a scale U such that for all scales V there exists a natural number n(V) such that for all V ∈ V there is a U -net, B, which has at most n(V) elements. Define W = st(U , U ), and let V be any scale of X. Let A be any W-net in V and B be the U -net with B ≤ n(V).
We claim that A ∖ B ≤ n(V). By contradiction assume A ∖ B > n(V). By maximality of B, given any As well, for any x j ∈ A∖B with x i ≠ x j we can find a y j with the same property above. If y i = y j , then {x i , x j } ⊂ st(U i , U ) which is a contradiction as A is a W-net in V . Now, take a collection of elements {x i } 1≤i≤n(V) of A ∖ B, and associate with each x i the unique y i ∈ B and form the collection However, by construction, given any y ∈ B, {x, y} ∉ U for all all U ∈ U . Otherwise, suppose there exists a y ∈ B with {x, y} ⊂ U. Since {y i } 1≤i≤n(V) = B, y = y i for some i and {x, x i } ⊂ st(U i , U ) which contradicts A being a W-net. Thus {x, y} ∉ U for all all U ∈ U . Therefore B is not maximal as x ∉ B. Hence, B is not a U -net which gives a contradiction. Thus A ∖ B ≤ n(V), and any W-net must have less than 2n(V) + 1 elements as W = st(U , U ). This satisfies the definition of bounded scale measure at scale W.
Our next definition is useful for working with scales as it gives a way to compare any scale V against the scale U where the U -nets are defined.
Proof. Let (X, LSS) have bounded scale measure at scale U , and let V be a scale of LSS. Let U and u(V) be defined as they are in the definition of bounded scale measure. Define W = st(U , U ). Take a U -net of V ∈ V and call it B. Then B ≤ u(V) since (X, LSS) has bounded scale measure. As U is a cover of X, is not a U -net which is a contradiction. Therefore for any U -net in V , B ≤ k(V). Then by definition, (X, LSS) has bounded scale measure at scale U .
Proof. Let (X, LSS) have bounded scale measure at scale U and let U ≺ V with V ∈ LSS. Let W be any other scale in LSS. Then by Theorem 2.2 any element W of W can be covered by n(U ) elements of U , call them U 1 , ..., U n(U ) . As U ≺ V, each The easiest example of large scale spaces with bounded scale measure is large scale spaces with bounded geometry. After all, bounded scale measure is meant to generalize bounded geometry. Bounded geometry is thought to be bounded scale measure at scale U where U is the cover of one point sets. Proof. Define U to be the trivial cover of X. Let V be any scale of X. Since (X, LSS) has bounded geometry, then for any V ∈ V there is an n ∈ N such that It turns out that if (X, LSS) has bounded scale measure and Y ⊆ X, then Y will have bounded scale measure too provided that we equip Y with the appropriate large scale structure: Definition 14. Let (X, LSS) be a large scale space and Y ⊆ X. The subspace large scale structure LSS ′ of Y is the collection of uniformly bounded families defined in the following way: Proof. Let (X, LSS) be a space with bounded scale measure at scale U , and let Y ⊆ X with (Y, LSS ′ ) the subspace large scale structure of (X, LSS).
Define V ′′ to be the trivial extension of V ′ to a scale of LSS ′ and u(V ′′ ) be the natural number from the definition of bounded scale measure for X. If V ′′ ⊂ X ∖ Y , then V ′′ must be a point and a U -net must also be a point. Otherwise V ′′ is an element of V ′ , and a U -net of V ′′ is also a U ′ -net of V ′ . This is because given any The following is a key result that distinguishes bounded scale measure from bounded geometry. A large scale space having bounded scale measure at some scale is a coarse invariant: We now show maximality. Assume by contradiction that there is some gives us that for any f (U )-net of V ∈ V has at most u(V) elements and thus (Y, LSS Y ) has bounded scale measure at scale f (U ). Conversely, let (Y, LSS Y ) be a space with bounded scale measure and let f ∶ X → Y be a coarse equivalence. By theorem 2.2 there exists a scale U in LSS Y such that for all scales V of LSS Y there exists a natural number k(V) so that for all V ∈ V, uniformly bounded cover of X. Take any uniformly bounded cover V of LSS X . Then f (V) is a cover of LSS Y . From the definition, there exists a collection of elements We now present some examples of spaces with bounded scale measure: Definition 15. A group G has a proper left invariant metric if the topology generated by the metric on the group is proper (i.e. the topological closure of a bounded set is compact) and additionally for any g, h, k ∈ G d(kg, kh) = d(g, h).
Remark 2.7. It's shown in [3] that any countable group has such a metric. Further, R n has the euclidean metric as a proper left invariant metric where R n is viewed as a group under addition. Proof. Let R > 0 and let U be the cover by open R balls. Let V be any other scale in the large scale structure. Since our large scale structure is induced from a metric, there exists S > 0 such that the scale consisting of S balls coarsens V. We will call this scale W. With the topology induced by the metric in mind, let W ∈ W be a specific element and consider the topological closure of W . This is a compact set due to the metric being proper and as U is an open cover, there is a finite subcollection and h be the points of the centers of W and W ′ respectively. Then, one can obtain a cover of W ′ by translating the centers of U 1 , ..., U k by the element hg −1 . This implies that every W ′ ∈ W can be covered by k elements of U . Namely, the balls hg −1 ⋅ U 1 , ..., hg −1 ⋅ U k . These are elements of U due to left-invariance. Hence any element of V can be covered by at most k elements of U and therefore (G, LSS) has bounded scale measure for the scale U of R balls by theorem 2.2.
From the proposition and the remark, we have the following result. A question we have is whether or not translation invariance is needed. This is to ask if any proper metric space has bounded scale measure at any cover of balls of radius R > 0. The following is an application of theorem 2.6.
Example 2.11. Let R have the large scale structure induced from the metric of absolute value. This will follow because it's known that Z with the absolute value metric is coarsely equivalent to R. Since Z with the large scale structure from the absolute value metric has bounded geometry (i.e. has bounded scale measure at the scale of the cover of one point sets), then R with the large scale structure induced from the absolute value metric also has bounded scale measure. It's important to note here that while R does not have bounded geometry, it has bounded scale measure by the above corollary. As shown earlier in the paper, bounded geometry is not in general preserved by coarse equivalences. However, bounded scale measure is preserved by coarse equivalences.
Here is another example of a large scale space having bounded scale measure. However, if U ∈ LSS is a cover by balls of radius less than 1, then U is a cover of single points. By choosing V to be the cover by balls of radius 1, then each element of V is all of R hence no element of V can be covered by a finite amount of elements of U .
We present an example of a space that does not have bounded scale measure at any scale.
Consider the large scale structure induced by this ∞-metric. Since any metrizable large scale structure is countably generated by balls of radius 0, 1, 2, .. we only need to show that any cover by radius n balls fails to cover all elements of some scale V with a finite union of n balls and will thus fail to have bounded scale measure by theorem 2.2. Let U be a cover by n balls where n is a non-negative integer. Let V be the cover of n + 1 balls. We will show that for all natural numbers k there does not exist a In the case of n = 0, this happens because a 1-ball in each component contains an increasingly large number of points which implies that a 1-ball in X contains infinitely many points ⃗ x. In the case of n > 0 this is because each element of V has at most (n+1) i elements in its Z i component, each element of U has at most n i elements in its Z i component, and the number (n + 1) i − n i is unbounded as i gets large. Thus the cover V can't be covered by a finite number of elements of U .

Generalizations of Property A
Property A was defined by Guoliang Yu [6] to approach the Baum-Connes conjecture. It can be viewed as a weaker version of amenability and is a sufficient condition for many properties such as coarse embeddability into a Hilbert Space. In its original context, property A was defined for a uniformly discrete metric space in the following way.
Definition 16. Let X be a uniformly discrete metric space. X has property A if for every ǫ > 0 and R > 0 there exists a collection of finite subsets {A x } x∈X , A x ⊆ X × N for every x ∈ X, and a constant S > 0 such that These conditions make the sets A x and A y be almost equal if d(x, y) ≤ R and disjoint if d(x, y) ≥ 2S. From there, Hiroki Sako defined property A in [5] for uniformly locally finite coarse spaces. Our definition is inspired by [5]: Definition 17. Let X be a set and let LSS be a Large Scale structure on X with bounded geometry. (X, LSS) is said to have Property A if for every ǫ > 0 and every uniformly bounded family U ∈ LSS, there exists V ∈ LSS and a family of finite It can be shown that our definition of property A for large scale spaces with bounded geometry is "equivalent" in some sense to Sako's definition of property A for uniformly locally finite coarse spaces. We will state the theorem here. The interested reader can read a refresher on coarse spaces and the proof of the following theorem in the appendix at the end of this paper: Because of this theorem, we will drop the superscripts from the collections {A x } x∈X . We now show that our definition of property A is a coarse invariant.

From the construction of A x 's and bounded geometry it follows that
. Putting this all together we get and therefore (X, LSS X ) has property A.
The large scale definition of property A allows us to approach relationships with other coarse properties in a more topological way. As such, we will use this definition to show that finite asymptotic dimension implies the large scale definition of property A. Asymptotic Dimension can be thought of as a large scale geometric version of the covering dimension. One of the first useful definitions involved the concept of Rmultiplicity of a scale on a metric space.
Definition 18. For a scale U , the R-multiplicity of U on a metric space is the smallest positive integer n such that for every x ∈ X the ball B(x, R) intersects at most n elements of U .

This in turn inspired the following definition from [2]:
Definition 19. Suppose that X is a metric space. The asymptotic dimension of X is the smallest positive integer n such that for every R > 0 there exists a uniformly bounded cover U with R-multiplicity n + 1. It is denoted by asdim(X) = n.
From there, asymptotic dimension was generalized for large scale structures in [2].
Definition 20. Let (X, LSS) be a large scale space with large scale structure LSS. We say asdim(X, LSS) ≤ n if for every uniformly bounded cover U in LSS there exists a uniformly bounded cover V such that V is a coarsening of U with multiplicity at most n + 1 (i.e. each point x ∈ X is contained in at most n + 1 elements of V).
Since the trivial cover of X is uniformly bounded, this definition gives us that there exists a uniformly bounded cover B such that each x ∈ X is contained in at most n + 1 elements of B. It can be shown that asdim(X, LSS) ≤ n if LSS can be generated by a uniformly bounded family B such that the multiplicity of B is at most n + 1 [2]. As for the metric case, we will show that finite asymptotic dimension for large scale spaces implies the general large scale definition of property A. First we will introduce some notation and a lemma. Proof. Let z ∈ st(y, st n−1 (U )) be arbitrarily chosen. Then there exists U ∈ U such that z, y ∈ st(U, st n−2 (U )). Since y ∈ st(x, U ) there exists U ′ ∈ U with x, y ∈ U ′ . Therefore we have U ′ ∩ st(U, st n−2 (U )) ≠ ∅ and thus by the definition of stars and the above we have z ∈ st(U, st n−2 (U )) ⊂ st(U ′ , st(U , st n−2 (U ))). Putting this all together we then have which shows z ∈ st(x, st n (U )) and therefore st(y, st n−1 (U )) ⊆ st(x, st n (U )). For each x ∈ X define A x in the following way: Then each A x is finite due to the multiplicity of V against st(x, st n (U )) is at most k + 1 for all x ∈ X. It also follows that A x ⊆ st(x, W) × N and (x, 1) ∈ A x . All that remains is to check that the inequality in the definition of property A is satisfied. Let y ∈ st(x, U ). Since st({x} x∈X , st n (U )) ≺ V, there exists V ∈ V with st(x, st n (U )) ⊆ V . From the construction of A x for this fixed V ∈ V we have (z V , m) ∈ A x for 1 ≤ m ≤ n. Additionally, we have y ∈ V and from our lemma we have (z V , m) ∈ A y for 1 ≤ m ≤ n − 1. Therefore, A x ∩ A y ≥ n − 1. On the other hand, consider (z V , l) ∈ A x △ A y . Without loss of generality suppose On the other hand, we know that either V ∩ st(y, st l (U )) = ∅ or V ⊂ st(y, st l (U )). First suppose that V ∩ st(y, st l (U )) = ∅. From the previous lemma we have that st(x, st l (U )) ⊂ st(y, st l+1 (U )), so V ∩ st(y, st l+1 (U )) ≠ ∅. On the other hand, suppose V ⊂ st(y, st l (U )). Again, by the above lemma we have V ⊂ st(y, st l (U )) ⊆ st(x, st l+1 (U )). Therefore any (z V , l) can only be contained in A x ∖ A y for at most 2 distinct values of l. Each A x can only have at most k + 1 distinct z V 's so A x △ A y ≤ 2(2(k + 1) + 1). Putting both inequalities together we get that The numerator is independent of our choice of n, so we may choose an n great enough so that 4k+6 n−1 < ǫ to satisfy the inequality. Therefore the collection {A x } x∈X satisfies the requirements and (X, LSS) has property A.
At this juncture, we seek to construct a definition of property A that does not depend on the large scale structure having bounded geometry. We finally consider a generalization of property A that occurs in spaces with bounded scale measure.
If (X, LSS) has bounded scale measure at scale U , then it's useful to imagine the elements of U as "points". In our previous definition of property A, we needed our points in a specific A x to be "near" one another (A x ⊂ st(x, V) × N). To accomplish this property for our "points" U ∈ U , we will make use of a set called a horizon: Definition 21. The horizon of a set against a scale is defined in the following way: We now introduce the definition of property A for a large scale space with bounded scale measure: Definition 22. Let (X, LSS X ) be a large scale space with bounded scale measure. We say that (X, LSS) has property A at scale U if for all ǫ > 0 and for all V ∈ LSS X that is not the trivial cover, there exists a collection of finite subsets {A U } U ∈U with A U ⊆ U × N for all U ∈ U and W ∈ LSS X such that:

3.)
Remark 3.6. We note here that while each element of {A U } U ∈U is a collection of finite subsets of U , {A U } U ∈U itself may not contain finitely many points of X.
We also note here that if one has property A at scale E where E is the trivial cover, then we have the definition of property A that was discussed earlier in the paper.
Lastly, it's useful to think of having property A at scale U as having property A but consider the elements of U as "points" as opposed to elements of X. In doing this, we can contain complexity within elements of U and see if the the space has property A-like behavior at a larger scale.
We now show having property A at a scale is a coarse invariant. Proof. Let (X, LSS X ) have bounded scale measure at scale U X . Let (Y, LSS Y ) have bounded scale measure at scale U Y and let LSS Y have property A at scale U Y . By theorem 2.2, let n be so that any element of the scale f −1 (U Y ) is covered by at most n elements of U X ; likewise, let m be so that any element of the scale f (U X ) can be covered by at most m elements of U Y . Let V X be any scale in LSS X and let ǫ > 0 be given. Since f is a coarse equivalence both f (U X ) and f (V X ) are scales in LSS Y . Therefore st(f (V X ), f (U X )) is a uniformly bounded family in LSS Y and we will set is a uniformly bounded scale, it must be that for all U X ∈ U X there exists is a scale in X and therefore each element of this scale can be covered by n elements of U X from the definition of bounded scale measure. Thus, for constructing A U X we will only be using the m elements from U Y that we previously where U X i and B U Y j are from the chosen n and m elements respectively. As well, We must now show that our chosen A U X and W X satisfy the property A at scale U X . By construction, each A U X contains the element (U X , 1).
Next, we will now show that any A U X has the property that the definition of the star and the horizon. Observe that the set U Y j ∪ W Y ∪ U Y 1 is contained in an element of st(W Y , U Y ). We will call this element C. Then taking the inverse image one has Finally we must show that given any U X 1 , U X 2 ∈ U X with To begin, recall that for a chosen collection of m elements U Y j from U Y . Additionally, each U Y j contains at most n chosen f (U X i )'s and the original U X . This combined with the fact that both f (U X 1 ) and f (U X 2 ) may intersect m unique elements taken from the B U Y j each, we have However, since we have hor(U X 1 , V X ) ∩ hor(U X 2 , V X ) ≠ ∅ then for any of the counted B U Y k or B U Y l for U X 1 and U X 2 respectively it must be that We know there exists n elements of U X that cover f −1 (U Y * ). Thus there exists at least one element U X * ∈ U X with f (U X * )∩U Y * ≠ ∅. Therefore U X * ×{z} ∈ A U X 1 ∩A U X 2 . Additionally, we know that there exist m elements of U Y that cover f (U X * ) by construction. Thus, there are at most m elements of Finally, we choose a specific k and l so that Combining this altogether, we get and (X, LSS X ) has property A at scale U X .
Using the previous proof and that bounded scale measure is a coarse invariant, one has the immediate corollary: In closing, we would like to give a parting thought regarding possible connections between bounded scale measure and coarsely doubling spaces along with weakly paracompact spaces shown in [1]. Many of these ideas are similar or even equivalent in metric spaces. One wonders how they may compare in a more general setting. Additionally, it may be worthwhile to consider connections between property A at a scale and finite asymptotic dimension.

Appendix
We begin the appendix with a proof that a large scale continuous function f is a coarse equivalence if and only if f is a coarse embedding and is coarsely surjective: Proof. Assume that f ∶ X → Y is a coarse equivalence. First we will show that f is a coarse embedding. Let U be a uniformly bounded cover of Y and let g ∶ Y → X be the function such that g ○ f ∶ X → X is close to the identity id X . Then there exists a uniformly bounded family V of X such that for every x ∈ X there exists a V ∈ V with x ∈ V and g ○ f (x) ∈ V . Let U be a uniformly bounded family in Y . Since g is large scale continuous, then g(U ) = {g(U) ∶ U ∈ U } is a uniformly bounded family in X. We claim the uniformly bounded family W = st(g(U ), V) coarsens f −1 (U ). Let U ∈ U . Consider any x ∈ X such that f (x) = y ∈ U. From the definition of coarse equivalence, there exists a V y ∈ V such that {g ○ f (x), x} ⊆ V y . Therefore, V y for V y defined above. However, for y ∈ U we know {g(y), x} ∈ V y for all x ∈ X with f (x) = y, so V y ∩ g(U) ≠ ∅. Thus for all y ∈ U V y ⊆ st(g(U), V). Therefore f −1 (U) ⊆ st(g(U), V), and from the definition of a large scale structure we have f −1 (U ) is a uniformly bounded family hence f is a coarse embedding. Next we will show that f is coarsely surjective. From the definition of coarse equivalence, there exists a uniformly bounded cover U of Y such that for every y ∈ Y there exists a U ∈ U with {f ○ g(y), y} ⊆ U. Let y ∈ Y be arbitrary. Then there exists a U ∈ U with {f ○ g(y), y} ⊆ U. Therefore , U ), and f is coarsely surjective.
Conversely, suppose f is a large scale continuous map that is a coarse embedding and is coarsely surjective. First we will construct a large scale continuous function g ∶ Y → X. From coarse surjectivity, we know there exists a uniformly bounded family U in Y such that Y ⊆ st(f (X), U ). Thus, for every y ∈ Y there exists a U y ∈ U with U y ∩ f (X) ≠ ∅. Choose an x y ∈ X such that f (x y ) ∈ U y . Define g ∶ Y → X with g(y) = x y . We first show that f ○ g is close to id Y . Let y ∈ Y . From the construction of g, there exists U y ∈ U with y ∈ U y , and f ○ g(y) ∈ U y . Thus {f ○ g(y), y} ⊆ U y and thus f ○ g is close to id Y . Now we will show that g ○ f is close to id X . Let y ∈ Y . By coarse embeddability we know f −1 (U ) is a uniformly bounded family in X. Choose x ∈ X. We know from coarse surjectivity that there exists and g ○ f is close to id X . Finally we will show that g is large scale continuous. Let U be a uniformly bounded family of Y . Since f ○ g is close to id Y , then there exists a uniformly bounded family V of Y such that for every y ∈ Y there exists V ∈ V with {f ○ g(y), y} ⊆ V . Let U ∈ U . For every y ∈ U, there exists V ∈ V such that {f ○ g(y), y} ⊆ V . Therefore, for every When giving the following definition, Sako states that a metric space with a uniformly locally finite coarse structure is called a metric space with bounded geometry.
Definition 26. (Sako Definition) A uniformly locally finite coarse space (X, C) is said to have property A if for every positive number ǫ and every controlled set T ∈ C, there exists a controlled set S ∈ C and a subset A C ⊂ S × N such that Given the above proposition, an equivalent definition of property A for Large Scale structures with bounded geometry can be made. We provide our definition below to refresh the reader: Definition 27. Let X be a set and let LSS be a Large Scale structure on X with bounded geometry. (X, LSS) is said to have property A if for every ǫ > 0 and every uniformly bounded family U ∈ LSS, there exists V ∈ LSS and a family of finite subsets {A LSS x } of X × N such that We end the appendix with the claim that our definition of property A for bounded geometry is equivalent to Sako's definition of property A. Earlier in the paper, this was theorem 3.1: We will show that A C ⊂ S × N. Given (x, y, n) ∈ A C then (x, y) ∈ {x} × A LSS x or y ∈ A LSS x ⊂ st(x, V). Thus, there exists a V ∈ V such that x, y ∈ V . Therefore (x, y) ∈ V × V , and A C ⊂ S × N.
Next, we will show that for x ∈ X, A C x = {(y, n) ∈ X × N; (x, y, n) ∈ A} = A LSS x . Let x ∈ X and consider (y, n) ∈ A C x . By definition (y, n) ∈ A LSS x . Similarly, if (z, m) ∈ A LSS x , then (x, z, m) ∈ A C , and therefore (z, m) ∈ A C x , so A C x = A LSS x By definition of A LSS x , and A C we get • For x ∈ X, A C x = {(y, n) ∈ X × N (x, y, n) ∈ A C } is finite The third property also follows easily as if (x, y) ∈ T , then there exists U ∈ U with x, y ∈ U. Thus, y ∈ st(x, U ), so we have and (X, C) has property A. Conversly, assume that (X, C) has Property A. Then we will show that (X, LSS) has property A, where LSS is the large scale structure induced by C. Let ǫ > 0 and U be a uniformly bounded cover. Thus there exists a controlled set T ∈ C such that ⋃ U ∈U U × U ⊂ T . Let S be the controlled set and A ⊂ S × N satisfying property A for ǫ and T . Define V = {V x } x∈X where V x = {y ∈ X; (y, n) ∈ A x , n ∈ N}. It needs to be shown that V is uniformly bounded: given (y, z) ∈ ⋃ V ∈V V ×V . Then, for some x ∈ X, (x, y, n), (x, z, m) ∈ A C ⊂ S × N for some n, m ∈ N. In addition, (x, y), (x, z) ∈ S meaning (y, x) ∈ S −1 and thus (y, z) ∈ S −1 ○ S which is a controlled set. Since (y, z) was chosen arbitrarily, we have that V is uniformly bounded. Note that st(x, V) = {y ∈ X; (y, n) ∈ A C z and (x, m) ∈ A C z for z ∈ X}. Since (x, 1) ∈ A C x , then A C x ⊂ st(x, V). Setting A LSS x = A C x one can show: x are true. Finally, if y ∈ st(x, U ), then there exists U ∈ U such that x, y ∈ U which means (x, y) ∈ ⋃ U ∈U U × U ⊂ T and the following holds: Thus (X, LSS) has property A.