Conditional Measures and Conditional Expectation; Rohlin’S Disintegration Theorem

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2012.32.2565 DYNAMICAL SYSTEMS Volume 32, Number 7, July 2012 pp. 2565–2582

CONDITIONAL MEASURES AND CONDITIONAL EXPECTATION; ROHLIN’S DISINTEGRATION THEOREM

David Simmons

Department of Mathematics University of North Texas P.O. Box 311430 Denton, TX 76203-1430, USA

Abstract. The purpose of this paper is to give a clean formulation and proof of Rohlin’s Disintegration Theorem [7]. Another (possible) proof can be found in [6]. Note also that our statement of Rohlin’s Disintegration Theorem (The- orem 2.1) is more general than the statement in either [7] or [6] in that X is allowed to be any universally measurable space, and Y is allowed to be any subspace of standard Borel space. Sections 1 - 4 contain the statement and proof of Rohlin’s Theorem. Sec- tions 5 - 7 give a generalization of Rohlin’s Theorem to the category of σ-ﬁnite measure spaces with absolutely continuous morphisms. Section 8 gives a less general but more powerful version of Rohlin’s Theorem in the category of smooth measures on C1 manifolds. Section 9 is an appendix which contains proofs of facts used throughout the paper.

1. Notation. We begin with the deﬁnition of the standard concept of a system of conditional measures, also known as a disintegration: Deﬁnition 1.1. Let (X, µ) be a probability space, Y a measurable space, and π : X → Y a measurable function. A system of conditional measures of µ with respect to (X, π, Y ) is a collection of measures (µy)y∈Y such that −1 i) For each y ∈ Y , µy is a measure on π (X). For µb-almost every y ∈ Y , µy is a probability measure. ii) The measures (µy)y∈Y satisfy the law of total probability Z µ(B) = µπ−1(y)(B)dµb(y) (1)

−1 for every event B of X. (Here and throughout this paper µb := µ ◦ π .) Note that we are implicitly assuming that the map y 7→ µy(B) is µb-measurable; we must be careful to prove this claim. The proof that we will give of Rohlin’s disintegration theorem is probabilistic; in particular, we will use the following notations motivated by a probabilistic point of view:

2000 Mathematics Subject Classiﬁcation. Primary: 28A50, 28C15; Secondary: 28C05. Key words and phrases. Disintegration, conditional measures, linear functionals, diﬀerential forms.

2565 2566 DAVID SIMMONS

Notation. Let (X, µ) be a probability space, A a µ-measurable subset of X with µ(A) > 0. We write µ(B ∩ A) P (X ∈ B X ∈ A) := µ (B) := µ A µ(A) Z Eµ(ψ(X ) X ∈ A) := ψ(x)dµA(x)

To prove the existence of systems of conditional measures, we will use a related concept which depends on topology: Deﬁnition 1.2. Let (X, µ) be a topological probability space, Y a metric space, and π : X → Y a measurable function. (π need not be continuous.) Let y ∈ Y . Then the topological conditional measure of µ with respect to (X, π, y, Y ) is the weak-* limit

µy := lim µπ−1(B(y,ε)) (2) ε→0 if it exists and is supported entirely on π−1(y). (The measures on the right hand side are defined by Notation1.) This definition has the advantage of being specific: for each y ∈ Y , there is at most one measure on π−1(y) which can be called the conditional probability of µ on π−1(y). Its disadvantage is that the context of the definition is less general: X is required to be a topological space and Y is required to be a metric space. We recall the following standard definitions:

Definition 1.3. Standard Borel space is the Cantor space 2N with its Borel σ- algebra; the Borel isomorphism theorem states that any uncountable Polish space with its Borel σ-algebra is Borel isomorphic to standard Borel space. Definition 1.4. A universally measurable space is a measurable space X such that there is an isomorphic embedding iX of X into standard Borel space, such that for every Borel measure µ on standard Borel space, iX (X) is in the completion of µ. Definition 1.5. A metric space X is an ultrametric space if it satisfies the ultrametric triangle inequality d(x, z) ≤ max(d(x, y), d(y, z)) for all x, y, z ∈ X.

2. Statement of Rohlin’s Disintegration Theorem. We will prove two ver- sions of Rohlin’s Theorem; the ﬁrst, which is a strengthening of the version given in [7], is an entirely measure-theoretic formulation, whereas the second, which appears to be new, involves topology. Theorems 2.1 and 2.2 correspond to Deﬁnitions 1.1 and 1.2, respectively. Theorem 2.1 (Rohlin’s Disintegration Theorem). Let X be a universally measurable space, let Y be a measurable space such that there exists a measurable injective map from Y into standard Borel space, and let µ be a Borel probability measure on X. Let π : X → Y be measurable. Then there exists a system of conditional measures (µy)y∈Y of µ with respect to (X, π, Y ). They are unique in the sense that if (νy)y∈Y is any other system of conditional measures, then µy = νy for µb-almost every y ∈ Y . ROHLIN’S DISINTEGRATION THEOREM 2567

Theorem 2.2. Let (X, µ) be a compact metric probability space, let Y be a locally compact separable ultrametric space or a separable Riemannian manifold. Let π : X → Y be measurable. Then for µb-almost every y ∈ Y , the topological conditional measure of µ with respect to (X, π, y, Y ) exists as in Deﬁnition 1.2. Furthermore the collection of measures (µy)y∈Y is a system of conditional measures as in Deﬁnition 1.1. (If µy does not exist, set µy = 0.) The proof will be divided into 2 parts: deducing Theorem 2.1 from Theorem 2.2, and proving Theorem 2.2.

3. Proof of Rohlin’s Theorem: Theorem 2.2 → Theorem 2.1. Let X0 = 2N 0 be standard Borel space, and let iX : X → X be the inclusion guaranteed by the 0 −1 0 0 universal measurability of X. Let µ = µ ◦ iX ; µ is a probability measure on X . Then (X0, µ0) is a compact metric probability space. 0 Let iY be a measurable injective map from Y into the Cantor space Y := 2N equipped with the Borel σ-algebra. Note that Y 0 is a locally compact separable ultrametric space. By [[8] 3.2.3 p.92], the map π admits a Borel measurable extension π0 : X0 → Y 0. Note that there is no reason to suppose that π0 is continuous. Thus we have satisfied the hypotheses of Theorem 2.2 for (X0, µ0, π0,Y 0). (If 0 X and Y are standard Borel, we are done with existence.) Let (µy0 )y0∈Y 0 be a system of conditional measures of µ0 with respect to (X0, π0,Y 0). For each y ∈ Y , let µ = (µ0 i (X)) ◦ (i−1)−1 if µ0 is supported on i (X), and µ = 0 y iY (y) X X iY (y) X y otherwise. Note that this makes sense since iX (X) is universally measurable. We claim that (µy)y∈Y is a system of conditional measures of µ with respect to (X, π, Y ). 0 0 First, note that since π is an extension of π, then iY ◦ π = π ◦ iX , and thus 0 −1 µb = µb ◦ iY . For all y ∈ Y , µ0 is a measure on (π0)−1(i (y)). If µ0 (X0 \ i (X)) > 0, iY (y) Y iY (y) X then µ = 0 is a measure on π−1(y). If µ0 (X0 \ i (X)) = 0, then µ = y iY (y) X y (µ0 i (X)) ◦ (i−1)−1 is a measure supported on i−1((π0)−1(i (y)) ∩ i (X)), iY (y) X X X Y X −1 which by the injectivity of iY is equal to π (y). Furthermore, in this case we have µ0 = µ ◦ i−1. If additionally µ0 is a probability measure, then µ is a iY (y) y X iY (y) y probability measure. 0 0 0 0 0 0 Now for µb -almost every y ∈ Y , µy0 is a probability measure, and µy0 (X \ 0 0 iX (X)) = 0. (The second claim follows from (1) applied to the formula µ (X \ 0 iX (X)) = µ(∅) = 0.) Thus for µ-almost every y ∈ Y , µ is a probability b iY (y) measure, and µ0 (X0 \ i (X)) = 0. By the preceding paragraph, we see that for iY (y) X −1 every y ∈ Y , µy is a measure on π (y), and for µb-almost every y ∈ Y , µy is a probability measure and µ0 = µ ◦ i−1. Thus condition (i) of Definition 1.1 is iY (y) y X satisfied. To prove condition (ii), fix B ⊆ X measurable. Since iX is an embedding, there 0 0 −1 0 exists B ⊆ X Borel such that B = iX (B ). Now for µb-almost every y ∈ Y , −1 0 −1 0 µiY (y) = µy ◦ iX and therefore µiY (y)(B ) = µy ◦ iX (B ) = µy(B). Thus the function y 7→ µy(B) is equal µb-almost everywhere to the composition of iY with the 0 0 map y 7→ µy0 (B ), and is therefore µb-measurable. Finally, note that µ0(B0) = µ(B). Applying (1), we see that µ(B) = R 0 0 0 0 R 0 0 R −1 0 R µ 0 (B )dµ (y ) = µ (B )dµ(y) = µy ◦ i (B )dµ(y) = µy(B)dµ(y). Thus y b iY (y) b X b b (1) is satisfied for (µy)y∈Y , which is therefore a system of conditional measures of µ with respect to (X, π, Y ). 2568 DAVID SIMMONS

It remains to show uniqueness. This actually follows from much weaker as- sumptions: from now until the end of this proof, rather than assuming that X is universally measurable and that there exists a measurable injective map from Y into standard Borel space, we will assume only that X and Y are measurable spaces, and that the σ-algebra of measurable subsets of X is countably generated. (This follows from the fact that X is universally measurable, since 2N is separable and any subset of a separable measurable space is separable.) The ﬁrst step will be to prove uniqueness for each individual event. The separa- bility of X will allow us to extend to the uniqueness stated in the theorem. We will need a lemma:

Lemma 3.1. If (µy)y∈Y is a system of conditional measures of µ with respect to (X, π, Y ), then for all measurable events S ⊆ Y and B ⊆ X, Z −1 µ(π (S) ∩ B) = µy(B)dµb(y). (3) S −1 Proof. The result follows directly from (1) and is left to the reader. Hint: µy(π (S) ∩ B) = χS(y)µy(B), which follows from the fact that µy is supported entirely on π−1(y). Corollary 1. Systems of conditional measures are unique in the sense that if (µy)y∈Y and (νy)y∈Y are two systems of conditional measures for the same measure µ, then for every event B of X and for µb-almost every y ∈ Y , µy(B) = νy(B). (Note the order of the quantifiers.) Proof. By Lemma 3.1, Z Z µy(B)dµb(y) = νy(B)dµb(y) S S for every measurable S ⊆ Y . Let S1 = {y ∈ Y : µy(B) < νy(B)} and S2 = {y ∈ Y : µy(B) > νy(B)}. If µ(S1) > 0, then the right hand side would be bigger, thus µ(S1) = 0. Similarly, µ(S2) = 0. Thus µy(B) = νy(B) for µb-almost every y ∈ Y . This corollary gives us the desired uniqueness for individual events. However, when we reverse the order of quantifiers, we can only guarantee that µy(B) = νy(B) for a countable collection of events B. Now we use the fact that the σ-algebra of X is countably generated; let (Bn)n∈N be a generating sequence. Then the collection of finite intersections (∩ B ) is also countable. Thus for µ-almost every n∈F n F ⊆N b #(F )<∞ y ∈ Y , µy(B) = νy(B) for each B in this collection. By [[2] 1.6.2 p.45], this implies that for µb-almost every y ∈ Y , µy(B) = νy(B) for every event B of X, i.e. µy = νy for for µb-almost every y ∈ Y . 4. Proof of Rohlin’s Theorem: Theorem 2.2. The heart of the proof is con- tained in the following lemma:

Lemma 4.1. If ψ : X → R is integrable, then the function

y 7→ Eµ(ψ(X ) π(X ) = y) := lim Eµ(ψ(X ) π(X ) ∈ B(y, ε)) (4) ε→0 is well-deﬁned for µb-almost every y ∈ Y , and is a Radon-Nikodym derivative of −1 (ψµ) ◦ π against µb. ROHLIN’S DISINTEGRATION THEOREM 2569

For the remainder of this section, we will take (4) as a definition of the topological conditional expected value of ψ with respect to (X, µ, π, y, Y ). The reason that this is not a good definition in general is that it leads to counterintuitive results. For example, if ψ = χπ−1(y), then Eµ(ψ(X ) π(X ) = y) = 0, yet for every value of x for which π(x) = y, ψ(x) = 1. For continuous functions this kind of thing doesn’t happen, which is why it was necessary to use weak-* convergence in Definition 1.2. Thus what the lemma really says is that conditional expected values exist almost everywhere, and if you know what their values are when they do exist, you can reconstruct the expected value of ψ on any event which depends only on π(X ). Proof of Lemma 4.1: Note that (ψµ) ◦ π−1(B(y, ε)) Eµ(ψ(X ) π(X ) ∈ B(y, ε)) = µb(B(y, ε)) Thus this lemma is really the Lebesgue differentiation theorem (Theorem 9.1) ap- −1 plied to the space Y and the measures (ψµ) ◦ π and µb. Only a few technical points stand between this lemma and the full Theorem 2.2. The first is the distinction between conditional expectation and conditional measure. Ideally, the conditional measures would be a collection of measures (µy)y∈Y such that for every integrable ψ : X → R and for every y ∈ Y , Z ψdµy = Eµ(ψ(X ) π(X ) = y). (5)

However, this is not possible, since the map A 7→ Eµ(χA(X ) π(X ) = y) is not −1 1 countably additive. (It is ﬁnitely additive.) As an example, if An = π (B(y, n )), then lim Eµ(χA (X ) π(X ) = y) = 1 6= 0 = Eµ lim χA (X ) π(X ) = y n→∞ n n→∞ n

The problem is that the expression Eµ(ψ((X )) π(X ) = y) is itself defined in terms of a limit, so the inequality above is another way of saying that limits don’t necessarily commute. This suggests that the solution is to force one of the limits to be uniform. Another issue that comes up is the issue of null sets. For each integrable ψ, there is a null set Nψ outside of which Eµ(ψ(X ) π(X ) = y) is well defined. If we want (5) to hold for a certain set of points S ⊆ Y and a certain class of functions Ψ ⊆ RX , then the larger the set of functions is, the smaller the set of points can be. Specifically, S ⊆ X \∪ψ∈ΨNψ. Thus if Ψ is uncountable (for example all measurable functions), there is no guarantee that S is nonempty. If we think about it a little, these problems really have the same root. The counterexample given above would not be a problem if we were allowed to ignore the point y, since it is a null set. The problem is that the null sets add up, we need some way to make the number of terms in our union countable. What countable set of functions should we use? Based on the comments following Lemma 4.1, it would make sense to use only continuous functions. One nice fact about the class of continuous functions is that it is separable; i.e. there exists a countable collection of continuous functions which is dense in the uniform topology. (To see this, note that by Urysohn’s metrization theorem [[9] 23.1 p.166] X can be embedded in the Hilbert cube [0, 1]N, which implies that there is a countable collection of continuous functions which separate points [i.e. the projection maps]. 2570 DAVID SIMMONS

Let Ψ be the Q-algebra generated by these functions plus the function which is identically one. Then Ψ is countable and separates points. The uniform closure of Ψ is an R-algebra which contains the constants and separates points, and so by the Stone-Weierstrass theorem [[9] 44.7 p.292] is equal to C(X). Thus Ψ is a countable dense subset of C(X).) This is excellent, since it brings in the uniform limits necessary to make the limits commute. Now let Ψ be our countable dense set, and then N = ∪ψ∈ΨNψ is a null set. The ﬁrst thing that we want to show is that for all ψ ∈ C(X) = Ψ, then Nψ ⊆ N. (The closure denoted is the uniform closure.) To see this, pick a sequence ψn ∈ Ψ which tends to ψ uniformly. Then for all y ∈ Y \ N, then Eµ(ψn(X ) π(X ) = y) exists for all n ∈ N. Now

Eµ(ψ(X ) π(X ) = y) = lim lim Eµ(ψn(X ) π(X ) ∈ B(y, ε)) ε→0 n→∞

= lim lim Eµ(ψn(X ) π(X ) ∈ B(y, ε)) n→∞ ε→0

= lim Eµ(ψn(X ) π(X ) = y); n→∞ (The exchange of limits is justified because the convergence is uniform with respect to n.) In particular, Eµ(ψ(X ) π(X ) = y) exists, so y ∈ Y \ Nψ. By taking the contrapositive we see that Nψ ⊆ N. Fixing y ∈ Y \ N, we see that for every ψ ∈ C(X), the limit (4) exists. By [[1] p.77, paragraph 1], this implies that the measures µπ−1(B(y,ε)) converge in the weak-* topology to a limit measure, which we shall call µy := limε→0 µπ−1(B(y,ε)), following the notation of Definition 1.2. Note however that we have not yet proven −1 that µy is supported entirely on π (y). (We do know that it is a probability measure, since X is compact.) The next thing we want to show is that the collection of measures (µy)y∈Y \N satisfies the Law of Total Probability. Ideally, this should follow directly from the −1 fact that (4) is a Radon-Nikodym derivative of (ψµ) ◦ π against µb. As stated earlier, this fact allows global information to be reconstructed from local information, and that is exactly what the Law of Total Probability is about. Again, however, we must worry about technicalities. First, let’s see exactly what Lemma 4.1 buys us. If ψ is any continuous function, R then by integrating both sides of (2), we get that ψdµy is equal to (4). Thus the R function y 7→ ψdµy is well-defined for µb-almost every y ∈ Y , is µb-measurable, and −1 is a Radon-Nikodym derivative of (ψµ) ◦ π against µb. According to the definition of the Radon-Nikodym derivative, this means that for any measurable set S ⊆ Y , Z Z Z ψdµy dµb(y) = ψdµ (6) S π−1(S) R where the left hand side is taken to include the assumption that the map y 7→ ψdµy is µb-measurable. The next step requires a bit of carefulness. We want to generalize (6) to all bounded measurable ψ. Let BM(X) be the set of bounded measurable functions from X to R. We know that C(X) ⊆ BM(X) is a dense subset, if BM(X) is given the topology of monotone convergence (Lebesgue-Hausdorff Theorem, [[8] 3.1.36 p.91]). Thus it suffices to show that the set of all ψ ∈ BM(X) which satisfy (6) is closed in the topology of monotone convergence. This follows from three applications of the Monotone Convergence Theorem, applied to the finite measures µ, µ, and µy. R R b Note that if ψn−→ψ monotonically, then ψndµy−→ ψdµy monotonically. Thus if n n ROHLIN’S DISINTEGRATION THEOREM 2571

R R y 7→ ψndµy is µb-measurable for all n ∈ N, then y 7→ ψdµy is µb-measurable as well. Thus, (6) is true for all bounded measurable ψ. In particular, if ψ = χB, then (6) simplifies to (3). If furthermore S = Y , then (3) simplifies to (1). It remains to show that for µb-almost every y ∈ Y , the measure µy is supported entirely on π−1(y). (This is a condition of both definitions 1.1 and 1.2.) This is one step in the proof that seems somewhat counterintuitive; it seems like −1 the fact that µy is supported entirely on π (y) should follow directly from the fact that the defining equation for µy converges in the first place. In fact, this would be true if we knew that π were continuous rather than just measurable.1 If π is measurable there are counterexamples. For example suppose that X := [0, 1]2 Y := [0, 1] ( y if y 6= .5 π(x, y) := 0 if y = .5

−1 and µ is Lebesgue measure on X. Then µ.5 is supported on π (0) rather than on π−1(.5). Thus we are forced to resort to a different argument. Note that saying that µy −1 −1 is supported entirely on π (y) is the same as saying that µy ◦ π = δy, where δy is a point mass at y. We will prove this equality setwise and then reverse the order of quantifiers. Taking (3) and substituting B = π−1(C), we get Z Z −1 χC (y)dµb(y) = µy ◦ π (C)dµb(y) S S for all measurable sets C,S ⊆ Y . By an argument similar to the proof of Corollary 1, we see that for every measurable set C ⊆ Y and for µb-almost every y ∈ Y , we have −1 δy(C) = χC (y) = µy ◦ π (C). (7) By taking a countable dense collection of Cs we can reverse the order of quantifiers, noting that the collection of all Cs for which (7) holds is closed under monotone convergence. Finally we plug in C = {y}; thus for µb-almost every y ∈ Y , −1 µy(π (y)) = 1.

5. Generalization to σ-finite measure spaces: Motivation. Now that Rohlin’s Theorem has been proven, we will state and prove a generalization to the category of σ-finite measure spaces with absolutely continuous morphisms. The first question to ask: why generalize to σ-finite measure spaces? So far the interpretation of Rohlin’s theorem has been entirely probabilistic, but it is not clear how σ-finite measures are related to probability. If µ is a nonzero finite measure, µ(B) it makes sense to define Pµ(X ∈ B) = µ(X) and this gives a probability measure. However, if µ is an infinite σ-finite measure, then the normalization constant µ(X) is ∞, and the preceding formula makes no sense. Nevertheless, we can still come

1 If π is continuous, then π∗ : M(X) → M(Y ) is continuous in the weak-* topology, thus for every y ∈ Y , −1 −1 µy ◦ π = lim µ −1 ◦ π = lim µB(y,ε) = δy. ε→0 π (B(y,ε)) ε→0 b 2572 DAVID SIMMONS up with examples of σ-ﬁnite measures out of which probability measures naturally arise:

2 Example 5.1. Let X = R , Y = R, π = πy : X → Y , and let µ be given by −(x−y)2/2 R ∞ R ∞ the density ρ(x, y)dxdy := e dxdy. Since −∞ −∞ ρ(x, y)dxdy = ∞, it follows that µ is an infinite σ-finite measure. However, we will see in Section8 that −x2 µπ−1(0) is given by the density e dx, which is finite and in fact normalizes to 2 Gauss measure. Thus it makes sense to say P (X ∈ B Y = 0) = √1 R e−x /2dx. µ 2π B The easiest way to understand this is under the Bayesian understanding of probability, i.e. probability as degrees of belief. Under this interpretation, we can understand a σ-finite measure as a person who doesn’t have enough information even to say that particular events have fixed probabilities. Next, the person is told that Y = 0. This does not give him enough information to say precisely what X is, but he now has a good enough grip on the situation to be able to symbolize his understanding as a probability measure.

3 Example 5.2. Let X = R , Z = R, π = πz : X → Z, and let µ be given by 2 the density ρ(x, y, z)dxdydz := e−(x−y+z) /2dxdydz. Again µ is an infinite σ-finite measure. The difference this time is that now µπ−1(0) is also an infinite σ-finite measure; in fact it is exactly the measure which we called µ in Example 5.1. The point of this example is that once we admit that σ-finite measures have probabilistic significance, it makes sense to apply Rohlin’s Theorem in a context where there are no finite measures to be seen. Going back to the previous discussion, the way we can understand Example 5.2 is that, as in the previous example, the original measure µ is a person who doesn’t have enough information even to say that particular events have fixed probabilities. He is told that Z = 0, but his knowledge of the particle’s location still can’t be represented as a probability measure. Finally he is additionally told that Y = 0, and then he can say, for example, that P (X < 0) = .5. Note further that in this example µb is not σ-finite. In fact, it is equal to ∞λ, where λ is Lebesgue measure. Thus, the measure µb is not really very useful; in particular, the denominator of (2) becomes ∞, making the limit zero. Clearly 0 should not be considered a conditional measure. The conclusion we can draw is that we need another measure ν to replace µb in the denominator of (2). ν should have the property that µb << ν. For example, in this case we could set ν = λ.

6. Generalization to σ-finite measure spaces: Statements of the generalized Rohlin Theorems. To state the generalized version of Rohlin’s Theorem, we will first need to generalize definitions 1.1 and 1.2. Definition 6.1. Let (X, µ) and (Y, ν) be measure spaces, and let π : X → Y be a measurable function. A system of conditional measures of µ with respect to (X, π, Y, ν) is a collection of measures (νy)y∈Y such that −1 i) For each y ∈ Y , νy is a (nonnegative) measure on π (y). ii) For each event B of X, the measures (νy)y∈Y satisfy the generalized law of total probability: Z µ(B) = νy(B)dν(y) (8) ROHLIN’S DISINTEGRATION THEOREM 2573

Deﬁnition 6.2. Let (X, µ) be a topological measure space, (Y, ν) a metric measure space, and π : X → Y a measurable function. (π need not be continuous.) Let y ∈ Y . Then the topological conditional measure of µ with respect to (X, π, y, Y, ν) is the weak-* limit −1 µ π (B(y, ε)) νy := lim (9) ε→0 ν(B(y, ε)) if it exists, is locally ﬁnite, and is supported entirely on π−1(y).

Note that in both of these definitions we used the symbol νy in place of µy. This allows us to use these formulas simultaneously with the originals (1) and (2) in the proof of Theorem 6.4 without any abuse of notation. If ν = µb and is a probability measure, then these definitions are equivalent to the original definitions. Theorem 6.3. Let (X, µ) be a universally measurable σ-finite measure space and let (Y, ν) be a σ-finite measure space such that there exists a measurable injective map from Y into standard Borel space. Let π : X → Y be measurable, and suppose that µb << ν. Then there exists a system of conditional measures (νy)y∈Y of µ with respect to (X, π, Y, ν). For ν-almost every y ∈ Y , νy is a σ-finite measure. The conditional measures are unique in the sense that if (γy)y∈Y is any other system of conditional measures, then νy = γy for ν-almost every y ∈ Y . Theorem 6.4. Let (X, µ) and (Y, ν) be locally compact locally finite separable metric measure spaces; also assume that Y is either an ultrametric space or a Rie- mannian manifold. Let π : X → Y be measurable, and suppose that µb << ν. Then for ν-almost every y ∈ Y , the topological conditional measure of µ with respect to (X, π, y, Y, ν) exists as in Definition 6.2. Furthermore the collection of measures (νy)y∈Y is a system of conditional measures as in Definition 6.1.

7. Generalization to σ-finite measure spaces: Proofs of the generalized Rohlin Theorems. The proofs will follow the same format as the proofs of The- orems 2.1 and 2.2. Theorem 6.3 will be proven first under the assumption that Theorem 6.4 is known, and then Theorem 6.4 will be proven. ` Proof of Theorem 6.3 using Theorem 6.4: Since X is σ-finite, let X = An, n∈N where each An has finite µ-measure. Each An is a measurable subset of a universally measurable space, and is therefore also universally measurable. For each n ∈ N, let in : An → 2N be an isomorphic embedding such that in(An) is universally 0 measurable. By gluing we obtain an isomorphic embedding iX : X → X := N × 2N 0 −1 such that iX (X) is universally measurable. Letting µ = µ ◦ iX , we see that (X0, µ0)is a locally compact locally finite separable ultrametric measure space. 0 Similarly, we obtain a measurable injective map iY : Y → Y := N × 2N with the 0 −1 0 0 property that ν := ν ◦ iY is locally finite. We see that (Y , ν ) is a locally compact locally finite separable ultrametric measure space. Again by [[8] 3.2.3 p.92], we can extend π to a Borel measurable map π0 : X0 → Y 0. Thus we have satisfied 0 0 0 0 0 0 the hypotheses of Theorem 6.4 for (X , µ , π ,Y , ν ). Let (νy0 )y0∈Y 0 be a system of conditional measures of µ0 with respect to (X0, π0,Y 0, ν0). The remainder of the proof of existence given for Theorem 2.1 is valid, if we 0 0 replace each occurence of µb by ν, µb by ν , µy by νy, and “probability” by “σ- finite”. (Recall that a locally finite measure on a locally compact separable metric space is σ-finite, so the measures coming from Definition 6.2 are necessarily σ-finite.) 2574 DAVID SIMMONS

To prove uniqueness, we shall first assume that µ is finite. In this case, R νy(X)dν(y) = µ(X) < ∞, so νy(X) < ∞ for ν-almost every y ∈ Y . Thus the proof of uniqueness given for Thoerem 2.1 still holds, with the same replacements as above. ` If µ is a σ-finite measure, again let X = An, where each An has finite n∈N µ-measure. Note that for every system of conditional measures (νy)y∈Y of µ with respect to (X, π, ν) and for each n ∈ N, then (νy An)y∈Y is a system of conditional measures of µ An with respect to (An, π, ν). Thus if (νy)y∈Y and (γy)y∈Y are two systems of conditional measures, then for each n ∈ N,(νy An)y∈Y and (γy An)y∈Y are both systems of conditional measures for (µ An, π, ν). Uniqueness for the finite case implies that for ν-almost every y ∈ Y , νy An = γy An. Fixing y ∈ Y and letting n ∈ N vary, we see that for ν-almost every y ∈ Y , then νy An = γy An for all n ∈ N. (The countability of N justifies this reversal of P P quantifiers.) For each such y ∈ Y , νy = νy An = γy An = γy. Thus n∈N n∈N νy = γy for ν-almost every y ∈ Y .

Now we must prove Theorem 6.4; we will use Theorem 2.2. Note that there are two different generalizations made from Theorem 2.2 to Theorem 6.4: the generalization from ν = µb to ν >> µb, and the generalization from compact spaces to locally compact spaces. We will deal with the former generalization first; i.e. first we will prove Theorem 6.4 in the case where X is compact, then we shall generalize.

Proof of Theorem 6.4 in the case where X is compact: Since µ is assumed to be locally finite, it follows that it is finite. Thus the hypotheses of Theorem 2.2 are µ satisfied for the normalized measure µX := µ(X) . It is left to the reader to verify that using µ instead of µX does not affect (1) and (2). The first thing that Theorem 2.2 tells us is that for µb-almost every y ∈ Y , the weak-* limit (2) exists and is entirely supported on π−1(y). We compare (2) with (9), and see that they differ by a factor of µ(B(y, ε)) f(y) := lim b . (10) ε→0 ν(B(y, ε)) By Theorem 9.1 (Lebesgue differentiation theorem), this limit exists and is finite for ν-almost every y ∈ Y . Furthermore the function f thus defined is a Radon-Nikodym derivative of µb against ν. Taking products, we see that (9) exists and equals f(y)µy for µb-almost every y ∈ Y . This is of course not enough; we need to show that it exists for ν-almost every y ∈ Y . This can be remedied by the following argument: Let F be the set of all y ∈ Y such that the conditional measure µy exists according to Definition 1.2. Then N := X \ F is a µ-nullset, so R f(y)dν(y) = µ(N) = 0. Thus f(y) = 0 for b N b ν-almost every y ∈ N. Since (2) is bounded, f(y) = 0 implies that (9) is zero. Thus (9) is zero for ν-almost every y ∈ N. (Note that this is true even though N is not necessarily a ν-nullset.) In conclusion, we have shown that for ν-almost every y ∈ Y ,(9) exists and is given by the following formula: ( f(y)µy if y∈ / N νy = (11) 0 if y ∈ N where µy is a conditional measure. ROHLIN’S DISINTEGRATION THEOREM 2575

To complete the proof we will need to show that for ν-almost every y ∈ Y , νy is finite and supported entirely on π−1(y), and that (8) holds for any event B of X. For the former claim, let y ∈ Y satisfy (11). If y ∈ N, then νy = 0 is trivially −1 finite and supported on π (y). On the other hand, if y ∈ F , then µy exists and satisfies Definition 1.2; i.e. it is probability and entirely supported on π−1(y). Now −1 if f(y) < ∞, then νy is also finite and supported on π (y). Since f(y) < ∞ for ν-almost every y ∈ Y , we are done. For the latter claim, we note that Theorem 2.2 implies (1). Thus it suffices to show that the right hand sides of (1) and (8) are equal, i.e. Z Z µy(B)dµb(y) = νy(B)dν(y) This is a direct computation based on (11), and is left to the reader. Proof of Theorem 6.4 in the general case: Lemma 7.1. If X is a locally compact separable metric space, then there exists an increasing sequence of compact sets Kn ⊆ X such that X = ∪n∈NKn, and such that for any compact set K ⊆ X, there exists N such that K ⊆ KN . Proof. Since X is locally compact, there is an open cover consisting of relatively compact open sets. By [[9] 16.11 p.112], X is Lindelöf,thus there is a countable subcover (Uj)j∈N. Let Kn := ∪j

Let (Kn)n∈N be as in Lemma 7.1. For each n ∈ N, the compactness of Kn implies that the quintuple (Kn, µ Kn, π, Y, ν) falls into the category of quintuples for which we have already proven that Theorem 6.4 applies. Denote the topological conditional measure of µ Kn with respect to (Kn, π, y, Y, ν) by νn,y, if it exists and satisﬁes the conditions of Deﬁnition 6.2.

Claim 7.2. Fix y ∈ Y , and suppose that for each n ∈ N, the conditional measure νn,y exists and satisﬁes the conditions of Deﬁnition 6.2. Then the sequence

(νn,y)n∈N is monotone increasing, and the limiting measure νy := limn→∞ νn,y is the topological conditional measure of µ with respect to (X, π, y, Y, ν).

Proof. The fact that (νn,y)n∈N is monotone increasing follows directly from the fact that (Kn)n∈N is a monotone increasing sequence of sets. By Lemma 9.2, the map

νy(A) := lim νn,y(A) n→∞ is a measure. We need to show that it is a locally ﬁnite measure, and that it is + equal to the weak-* limit (9). To this end, let ψ ∈ Cc (X); we claim that Z R −1 ψdµ π (B(y, ε)) ψdνy = lim < ∞. (12) ε→0 ν(B(y, ε))

Let N be large enough so that Supp(ψ) ⊆ KN . Then for all n ≥ N, the right hand R side can be calculated entirely on Kn, and is by deﬁnition equal to ψdνn,y < ∞. R Thus for all n ≥ N, ψdνn,y is independent of n. Taking the limit as n tends to R inﬁnity yields that the right hand side of (12) is equal to limn→∞ ψdνn,y. But by R (15), this is equal to ψdνy, proving (12). 2576 DAVID SIMMONS

For every x ∈ X, pick an open neighborhood U of x which is relatively compact. + By [[2] 7.1.8 p.199], there is a function ψ ∈ C (X) with χU ≤ ψ. Then νy(U) ≤ R c ψdνy < ∞. Thus νy is locally finite, as claimed. −1 It remains to show that νy is supported entirely on π (y). Fix y ∈ Y and let ψ := χX\π−1(y);(15) simplifies to −1 −1 νy(X \ π (y)) = lim νn,y(X \ π (y)) = 0. ε→0 / Note that the hypotheses of Lemma 7.2 are satisfied for ν-almost every y ∈ Y . Thus the first claim of Theorem 6.4 is proven, and it remains to show that (8) is satisfied for every event B of X. To see this, fix y ∈ Y , and let ψ := χB. Then (15) simplifies to νy(B) = limε→0 νn,y(B). Integrating both sides with respect to ν yields Z Z νy(B)dν(y) = lim νn,y(B)dν(y) n→∞ Z = lim νn,y(B)dν(y) n→∞

= lim µ(B ∩ Kn) n→∞ = µ(B) and we are done.

8. Application to Differentiable Manifolds. Next, we describe in more detail the specific case where X and Y are manifolds, and π is a smooth map: Theorem 8.1. Let X be an (m + n)-dimensional oriented C1 manifold, and let Y be an n-dimensional oriented C1 Riemannian manifold.2 Let π : X → Y be a C1 nonsingular map. Let µ and ν be nonnegative smooth measures on X and Y , respectively. Suppose that the densities of µ and ν are given by ω := dµ ∈ Γ(∧m+n T ∗X) λ := dν ∈ Γ(∧n T ∗Y ) where ω ≥ 0 and λ > 0 are measurable. Assume additionally that σ ∈ Γ(∧m T ∗X) is an m-form on X which satisfies ω = σ ∧ π∗λ (13) ∗ (By Proposition1, we know that such a σ exists and that iyσ is defined uniquely by ω and λ. It should also be clear that if π−1(y) is given the proper orientation, then ∗ iyσ ≥ 0.) −1 For each y ∈ Y , let νy be the smooth measure on π (y) corresponding to the density ∗ m ∗ −1 dνy := iyσ ∈ Γ(∧ T [π (y)]) −1 where iy : π (y) → X is the inclusion map. Then the collection of measures (νy)y∈Y is a system of conditional measures 1 according to Definition 6.1. If additionally σ is continuous and λ ∈ Lloc, then each measure νy is a conditional measure according to Definition 6.2.

2In place of assuming that X and Y are orientable, it may be assumed that the forms ω, λ, and σ deﬁned below are forms of odd type in the sense of [[3] Section 5, p.19-23] ROHLIN’S DISINTEGRATION THEOREM 2577

For example, this theorem can be used to calculate

1 ln(2) P X > .5 XY = = . µ 3 ln(3) We ﬁrst prove Theorem 8.1 in the exceptionally simple case where X = Rm+n, Y = Rn, and π is projection onto the last n coordinates: Proof of Theorem 8.1, Special Case: Write ∗ i~yσ(x1, . . . , xm) := f(x1, . . . , xm, y1, . . . , yn)dx1 ∧ ... dxm

λ(y1, . . . , yn) := g(y1, . . . , yn)dy1 ∧ ... dyn. An easy calculation shows that

ω(x1, . . . , xm, y1, . . . , yn)

= f(x1, . . . , xm, y1, . . . , yn)g(y1, . . . , yn)dx1 ∧ ... dxm ∧ dy1 ∧ ... dyn. Thus for every event B of X, by Fubini’s theorem Z µ(B) = f(x1, . . . , xm, y1, . . . , yn)g(y1, . . . , yn)dV(x, y) B Z "Z # = f(x1, . . . , xm, y1, . . . , yn)g(y1, . . . , yn)dV(x) dV(y) −1 Y i~y (B) Z "Z # = f(x1, . . . , xm, y1, . . . , yn)dV(x) g(y1, . . . , yn)dV(y) −1 Y i~y (B) Z = νy(B)dν(y).

Thus the collection of measures (νy)y∈Y is a system of conditional measures according to Definition 6.1. 1 It remains to prove (9), assuming that σ is continuous and that λ ∈ Lloc. Since the convergence is intended to be weak-*, we need to show that for every continuous m+n n function with compact support ψ : R → R,(12) holds. Since (ν~y)~y∈R is a system of conditional measures, the right hand side simplifies to Z ~ lim Eν ψdν ~ Y ∈ B(~y, ε) . ε→0 Y 1 R Note that this makes sense because λ ∈ Lloc. Writing Ψ(~y) := ψdν~y, we see that (12) simplifies to Ψ(~y) = lim Eν Ψ(Y~) Y~ ∈ B(~y, ε) (14) ε→0 Now the function Ψ is continuous; this follows from the continuity of ψ and of σ, and from the fact that ψ is compactly supported. By the definition of continuity this means that for every γ > 0 there exists an δ > 0 such that Y~ ∈ B(~y, δ) ⇒ |Ψ(Y~) − Ψ(~y)| ≤ γ. Fixing γ and δ, this implies that ~ ~ 0 < ε ≤ δ ⇒ |Eν (Ψ(Y) Y ∈ B(~y, ε)) − Ψ(~y)| ≤ γ. Adding the quantifiers back on, this is the same as saying that

lim Eν (Ψ(Y~) Y~ ∈ B(~y, ε)) = Ψ(~y) ε→0 2578 DAVID SIMMONS and we are done.

Proof of Theorem 8.1, General Case. The first thing we will do is to prove (8) for every event B of X. We first claim that it is sufficient to pick an open cover C, and to check that (8) holds whenever B ⊆ U ∈ C. To see this, note that X is Lindelöf,and thus C has a countable subcover (Un)n∈ . ` N Fix B ⊆ X measurable, and let Bn := B∩(Un \(∪i

9. Appendix. The version of the Lebesgue differentiation theorem which we have used is based on the theory developed in [[4] p.141-169]. Here it is: Theorem 9.1 (Lebesgue differentiation theorem). Let X be a locally compact separable ultrametric space or a separable Riemannian manifold. Let µ and ν be locally finite measures on X. Assume µ << ν. Then the function µ(B(x, ε)) f(x) := lim ε→0 ν(B(x, ε)) is well-defined for ν-almost every x ∈ X and is a Radon-Nikodym derivative of µ against ν. ROHLIN’S DISINTEGRATION THEOREM 2579

Proof. First, note that the case where X is a locally compact separable ultrametric space can be reduced to the case where X is a compact ultrametric space. To see this, let C be the cover of X consisting of all compact open balls. (Since X is ultrametric, open balls are closed, which is why the balls themselves are compact rather than relatively compact.) Since X is Lindel¨of, C has a countable subcover

(Un)n∈N. Next, write Vn := Un \ ∪i 0 : ∃x, y ∈ X 3 d(x, y) = r}. R cannot contain any sequence whose elements are all distinct and which does not tend to zero. To see this, suppose that (rn)n∈N is a sequence in R which does not tend to zero. Let xn, yn be such that d(xn, yn) = rn. Then there is a number ε > 0 and a subsequence of (rn)n∈N which is bounded below by ε. Since X is compact metric, there is a further subsequence of (xn, yn)n∈N which converges to a point (x, y). From now on (xn, yn)n∈N will refer to this sub-subsequence. We know that if n ∈ N is large enough, then d(x, xn) < ε and d(y, yn) < ε. Furthermore, d(xn, yn) = rn ≥ ε. The ultrametric inequality then implies that rn = d(xn, yn) = d(x, y) for all sufficiently large n ∈ N. This contradicts the fact that the (rn)n∈N are all distinct. Thus R consists of a sequence tending towards zero. (If R is finite then X is also finite, in which case the proof is trivial.) Index this sequence in decreasing order by

(rj)j∈N. We deﬁne a sequence of partitions (Pj)j∈N by letting Pj be the partition of X into open balls of radius rj. Note that each member of Pj is the union of some subfamily of Pj+1. Also, limj→∞ diam(Pj) = limj→∞ rj+1 = 0. Furthermore P1 is bounded and X is separable, so we have satisﬁed the hypotheses of [[4] 2.8.19 p.152]. Thus

V = {(x, S): ∃j ∈ N 3 x ∈ S ∈ Pj} = {(x, B(x, r)) : x ∈ X, : ∃j ∈ N 3 r = rj} = {(x, B(x, r)) : x ∈ X, 0 < r < ∞} is φ-Vitali. (The reason the last equality is true is that for any x ∈ X and for any 0 < r < ∞, there is some j ∈ N such that rj+1 < r ≤ rj. In this case, B(x, r) = B(x, rj).) In either case, we have satisfied the hypotheses of [[4] 2.9.1 p.152 (general as- sumptions to be used throughout Section 2.9)], noting that X is separable metric and therefore every locally finite measure is regular. In particular, the hypotheses of [[4] 2.9.5 p.154] are satisfied. The conclusion should be interpreted to mean that 2580 DAVID SIMMONS the derivate ψ(S) D(ψ, φ, V, x) : = (V ) lim S→x φ(S) ψ(S) = lim{ :(x, S) ∈ V, diam(S) < ε, φ(S) 6= 0} ε→0 φ(S) ψ(B(x, r)) = lim{ : diam(B(x, r)) < ε, φ(B(x, r)) 6= 0} ε→0 φ(B(x, r)) exists and is finite for φ-almost every x ∈ X. (See [[4] p.153] for the definition of derivate and [[4] p.151] for the definition of the notation (C) lim.) The statement that a limit of a parameterized collection sets (Rε)ε>0 exists should be interpreted to mean that the limits limε→0 sup(Rε) and limε→0 inf(Rε) exist and are equal. By the Sandwich Theorem, this implies that for any map f : (0, ε0) → ∪ε>0Rε such that f(ε) ∈ Rε, then limε→0 f(ε) exists and is equal ε ψ(B(x, 2 )) to this earlier limit. Thus if we take f(ε) := ε , then we find that for all φ(B(x, 2 )) x ∈ Supp(φ), ψ(B(x, ε )) D(ψ, φ, V, x) = lim 2 ε→0 ε φ(B(x, 2 )) ψ(B(x, ε)) = lim ε→0 φ(B(x, ε)) Thus this last line exists and is finite for φ-almost every x ∈ Supp(φ). Since φ(X \ Supp φ) = 0, this proves the well-definedness claim of Theorem 9.1. The proof that the limiting function x 7→ D(ψ, φ, V, x) is a Radon-Nikodym derivative of ψ against φ is given at [[4] 2.9.7 p.155]. Note that by [[4] 2.9.2 p.153], ψφ = ψ since ψ << φ. Next, we include a proposition about short exact sequences of vector spaces. The vector spaces can be over an arbitrary field, but for convenience of notation we assume they are all real vector spaces. For consistency with the geometric mean- ing we have considered the contravariant maps i∗ and π∗; however, corresponding statements could be made about the covariant maps i∗ and π∗. Proposition 1. Let U, V , and W be finite dimensional vector spaces, and let i : U → V and π : V → W form a short exact sequence. Suppose that dim(U) = m and dim(W ) = n, so that dim(V ) = m + n. Then if λ ∈ ∧n W ∗ is fixed, then (13) defines a law of proportionality between ω ∈ ∧m+n V ∗ and i∗σ ∈ ∧m U ∗. In other words, the set of all (ω, i∗σ) which satisfy (13) constitutes a one-dimensional linear subspace of ∧m+n V ∗ ⊕ ∧m U ∗.

Proof. Without loss of generality, we let U = Rm, V = Rm+n, W = Rn, i is inclusion, and π is projection. Fixing λ, we can write λ = ady1 ∧ dy2 · · · ∧ dyn for some a ∈ R. Next, we write each σ ∈ ∧m Rm+n as

σ =c1dx1 ∧ dx2 ∧ · · · ∧ dxm + c2dx1 ∧ dx2 ∧ · · · ∧ dxm−1 ∧ dy1

+ ... + c m+n dyn−m+1 ∧ dyn−m+2 ∧ · · · ∧ dyn ( m ) m+n where ci ∈ R for each i ∈ {1, 2,..., m }. (We have written the above formula as if n ≥ m; there is no essential difference if n < m.) Note that all terms except the first will become zero when the operation i∗ is applied, because each of them contains at least one factor which depends entirely ROHLIN’S DISINTEGRATION THEOREM 2581 in the y coordinates. Similarly, all terms except the first will become zero when the wedge product is taken with π∗λ, for the same reason. Thus

∗ i σ = c1dx1 ∧ dx2 ∧ · · · ∧ dxm ∗ σ ∧ π λ = ac1dx1 ∧ dx2 ∧ · · · ∧ dxm ∧ dy1 ∧ dy2 · · · ∧ dyn and since c1 can vary freely, we have established direct proportionality.

Corollary 2. In the context of Proposition1, then there is a natural isomorphism between ∧m+n V ∗ and ∧m U ∗ ⊗ ∧n W ∗.

∗ ∗ Proof. If λ is fixed, then Proposition1 establishes a linear map ×λ : i σ 7→ σ ∧ π λ from ∧m U ∗ to ∧m+n V ∗. Thus the map ×(i∗σ, λ) := σ ∧ π∗λ is well-defined and linear in the first coordinate. It is clear that × is linear in the second coordinate; thus it is a bilinear map. By the universal property of tensor products, there is a corresponding map from ∧m U ∗ ⊗ ∧n W ∗ to ∧m+n V ∗. It can easily be checked that this map is a surjection; since the domain and codomain have the same dimension (one), this implies that the map is a bijection.

Finally, a lemma from measure theory is required to complete an earlier argument.

Lemma 9.2. Let X be a measurable space and let µn be a monotonically increasing sequence of measures on X. Then the limiting function µ(A) := limn→∞ µn(A) is a measure; moreover for every nonnegative measurable function ψ : X → R, then

Z Z ψdµ = lim ψdµn (15) n→∞

Proof. Clearly µ(∅) = 0. Finite additivity is also clear. To show countable subad- ` ditivity, let A = Am. Then m∈N

µ(A) = lim µn(A) n→∞ X = lim µn(Am) n→∞ m∈N X ≤ lim µ(Am) n→∞ m∈N X = µ(Am) m∈N

Thus µ is a measure. To prove (15), we first note that if ψ is a characteristic function then this formula follows directly from the definition. If ψ is simple, then it is a positive linear combination of characteristic functions, and it is easy to show that (15) holds. Finally, suppose ψm increase monotonically to ψ, where ψm are 2582 DAVID SIMMONS simple. Then Z Z ψdµ = lim ψmdµ m→∞ Z = lim lim ψmdµn m→∞ n→∞ Z ≤ lim lim ψdµn m→∞ n→∞ Z = lim ψdµn n→∞ R R To prove the opposite inequality, note that µn ≤ µ, so ψdµn ≤ ψdµ. Taking the limit as n tends to infinity yields the desired result.

REFERENCES [1] H. Bergström,“Weak Convergence of Measures,” Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. [2] D. Cohn, “Measure Theory,” Reprint of the 1980 original, BirkhäuserBoston, Inc., Boston, MA, 1993. [3] G. de Rham, “Differentiable Manifolds. Forms, Currents, Harmonic Forms,” Translated from the French by F. R. Smith, With an introduction by S. S. Chern, Grundlehren der Mathe- matischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 266, Springer- Verlag, Berlin, 1984. [4] H. Federer, “Geometric Measure Theory,” Die Grundlehren der mathematischen Wis- senschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. [5] C. Hsiung, “A First Course in Differential Geometry,” Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1981. [6] D. Maharam, On the planar representation of a measurable subfield, in “Measure Theory, Oberwolfach 1983” (Oberwolfach, 1983), Lecture Notes in Math., 1089, Springer, Berlin, (1984), 47–57. [7] V. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp. [8] S. Srivastava, “A Course on Borel Sets,” Graduate Texts in Mathematics, 180, Springer- Verlag, New York, 1998. [9] S. Willard, “General Topology,” Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581], Dover Publications, Inc., Mineola, NY, 2004. Received May 2011; revised June 2011. E-mail address: [email protected]