Algebras and Maps

Conditional Measures and Algebras

5.4 Algebras and Maps

Moreover,

[x]_A =φ⁻¹(φ(x))

for x /∈ N, and μ^A_x = νφ(x) for some measurable map y → νy defined on a φ_∗μ-conull subset of Y. In fact we can take Y = M(X), φ(x) = μ^A_x , andνy =y.

This will be proved later; the conclusion described in Corollary 5.22 is depicted in Fig.5.3.

Fig. 5.3 Eachy∈Y determines an atomφ⁻¹(y) and its conditional measureνy

Lemma 5.23.If X is a compact metric space, and f ∈ L^∞(X), then the map

M(X)ν −→

fdν

is Borel measurable. In particular, for a Borel subset X of X, we have that M(X)is a Borel subset of M(X). Moreover, if φ:X →Y is a Borel measurable map between Borel subsets of compact metric spaces, then the induced mapφ_∗:M(X)→M(Y)is Borel measurable.

Proof. Starting with continuous functions, we know that

fdν depends continuously onν (by definition of the weak*-topology on M(X)). Arguing just as we did on p.139, this can be extended to show that

fdν depends measurably onν for all indicator functions of open sets, and thence to show that it does so for indicator functions of Borel measurable sets, and finally for anyf ∈L^∞. By definition and the argument above, it follows that

M(X) ={μ∈M(X)|μ(XX) = 0}

5.4 Algebras and Maps 147

is Borel measurable. Now letφ:X →Y be measurable and fixr∈R,ε >0 andf ∈C(Y). Then

Of,r,ε =

μ∈M(Y)|fdμ−r< ε is an open set inM(Y), and clearly

φ⁻_∗¹O_f,r,ε=

ν∈M(X)|f◦φdν−r< ε

is measurable in M(X). Since any open set in M(Y) can be written as a countable union of finite intersections of sets of the formO_f,r,εwithf chosen from a dense countable subset ofC(X),r∈Qandε∈Q, the lemma follows.

Proof of Corollary 5.22. Let A = σ({A₁, A₂, . . .}) be countably- generated. TakingY =M(X) (with the weak*-topology, so thatY is a com- pact metric space) andφ(x) =μ^A_x we can set ν_y =y and henceν_φ(x)=μ^A_x follows at once. LetX be aμ-conull set on which all the statements in The- orem5.14hold. We claim first that (perhaps after enlarging the complement ofX by null sets countably often),A =φ⁻¹BY. By Theorem5.14(2),

χA_i(x) =μ^A_x(Ai) (5.12) for almost everyx, and we may assume that this holds for allx∈X. Since {ν | ν(Ai) = 1} ∈BY, (5.12) shows that Ai∩X ∈ φ⁻¹BY and therefore A|X ⊆φ⁻¹BY.

For the reverse directionφ⁻¹BY ⊆A|X, it is sufficient to check this on sets of the formOf,r,εsince these generate the weak*-topology in a countable manner, and by Theorem5.14(1) the set

φ⁻¹ {ν| |

fdν−r|< ε}

={x| |

fdμ^A_x −r|< ε}

isA-measurable for anyf ∈C(X),r∈Randε >0. HenceA|X =φ⁻¹B. Sinceφ:X →Y satisfies A|X =φ⁻¹BY, and the Borelσ-algebraBY

ofY separates points, [x]_A =φ⁻¹(φ(x)) follows.

Corollary 5.24.Let φ: (X,BX, μ) → (Y,BY, ν) be a measure-preserving map between Borel probability spaces, and let A ⊆BY be a sub-σ-algebra.

Then

φ_∗μ^φ_x^−1A =ν_φ(x)^A forμ-almost everyx∈X.

Proof.First notice that for anyf ∈L¹(Y,BY, ν), Eν(fA)◦φis φ⁻¹A- measurable and

φ⁻¹A

E_ν(fA)◦φdμ=

A

E_ν(fA) dν

=

A

fdν

=

φ⁻¹A

f◦φdμ.

It follows that

Eν(fA)◦φ=Eμ(f◦φφ⁻¹A).

Thus forf ∈L^∞(Y,BY),

fdν_φ(x)^A =Eν(fA) (φ(x))

=Eμ(f ◦φφ⁻¹A)(x)

=

f◦φdμ^φ_x^−1A

=

fd

φ_∗μ^φ_x^−1A

(all almost everywhere). Using a dense countable subset ofC(X) completes

the proof.

In the next result we will work with measurability on Borel subsets of compact metric spaces, without reference to a particular measure.

Lemma 5.25.Let X, Y, Z be Borel subsets of compact metric spaces X,Y and Z respectively, and let φ_Z : X → Z and φ_Y : X → Y be measurable maps. Suppose that φ_Z is φ⁻_Y¹(BY)-measurable. Then there is a measurable mapψ:Y →Z with φ_Z(x) =ψ◦φ_Y(x)on X, as illustrated in Fig.5.4.

Fig. 5.4 The map constructed in Lemma5.25

This is related to Corollary 5.22, in that it allows us to draw the same conclusion that we can write the conditional measure μ^A_x as a measurable functionνy on the image spaceY wheneverA =φ⁻¹BY.

Proof of Lemma5.25.DefineA =φ⁻_Y¹(BY), which is countably-generated sinceBY is. Since the Borelσ-algebraBY ofY separates points,

[x]_A =φ⁻_Y¹(φY(x))

5.4 Algebras and Maps 149

forx∈X. By assumption, φZ isA-measurable, and henceφZ(x) =φZ(x) whenever [x]_A = [x]_A, or equivalently whenever φY(x) = φY(x). So we could define

ψ(y) =φZ(x)

whenever y = φY(x) for some x ∈ X, and use some fixed z0 ∈ Z to de- fine ψ(y) = z₀ for y ∈ Yφ_Y(X). However, it is not clear why this should define a measurable function, so instead we will defineψ:Y →Z by a lim- iting process. In order to do this we will cut the target space Z into small metric balls, and ensure that at each finite stage everything is appropriately measurable. For this we start with the additional assumption thatZ=Z is a compact metric space, and later show that this requirement can be removed.

Since Z is compact, there is a sequence (ξn) of finite Borel measurable partitions ofZ with the property that

σ(ξn)⊆σ(ξn+1)

forn1, and for which every element ofξn has diameter less than _n¹. We will define related partitionsξ^X_n andξ_n^Y ofX andY. The first of these is defined by taking pre-images as follows. SinceφZ isA-measurable we get, for anyP ∈ξn, a set

P^X=φ⁻_Z¹(P)∈A,

and hence a finite partitionξ_n^X={P^X |P ∈ξ_n} ⊆A ofX. This is illustrated in Fig.5.5.

Fig. 5.5 The partitionsξnonM(X) andξ_n^XonX

We next define the partition ξ_n^Y. Note that in the construction of ξ_n^Y care needs to be taken, since the set φY(X) is not assumed to be mea-

surable. Since A = φ⁻_Y¹BY, we can choose for every P^X ∈ ξ^X_n a set P^Y in BY with P^X =φ⁻_Y¹P^Y. Since the various sets P^X ∈ξ_n^X are disjoint, we can require the same disjointness for the sets P^Y. Indeed if sets P and Q in ξn are different, P^X = φ⁻_Z¹P = φ⁻_Y¹P, and Q^X = φ⁻_Z¹Q = φ⁻_Y¹Q^Y with Q^Y, P^Y ∈ BY not disjoint, then replacing Q^Y by Q^YP^Y will not change its properties but will ensure disjointness fromP^Y. Moreover, using a similar argument inductively we can also insist that P_n+1^Y ⊆ P_n^Y when- everP_n+1^X ⊆P_n^XandP_n+1^X ∈ξ_n+1^X ,P_n^X∈ξ^X_n. The setsP^Y ∈BY forP∈ξ_n^X, together with their common complementQ_n ∈BY, form a partitionξ_n^Y ofY. By construction,φ⁻¹Q_n=∅and Q_n+1⊇Q_n for alln1. Write

Q=

m1

Qm

and define ψ(y) to be some fixed element z0 ∈ Z for any y ∈ Q. We de- fine ψ on YQ as the limit of the sequence of functions ψn : YQ → Z defined as follows. For every P^Y ∈ ξ^Y_n, we choose some zP ∈ P ∈ ξn and define ψⁿ(y) = zP for y ∈ P^YQ. Clearly ψⁿ : YQ → Z is measurable, since the partitionξ_n^Y is measurable by construction. By construction of the partitions ξn, the diameters of the partition elementsP ∈ξn go to zero, so the functionψdefined by

ψ(y) = lim

n→∞ψ_n(y)

exists for ally ∈YQ; since it is a pointwise limit of a sequence of measur- able functions, the mapψis also measurable. Ify =φY(x) for somex∈X, then φZ(x) and ψn(y) will belong to the same element of ξn for all n, soφZ(x) =ψ(φY(x)) as required.

Assume now thatZ is only a Borel subset of a compact metric space Z.

The construction above can be used to define a measurable mapψ:Y →Z withφZ =ψ◦φY. Since

∅=φ⁻_Z¹ZZ

=φ⁻_Y¹

ψ⁻¹ZZ we may replaceψby the measurable map

ψ(y) =

ψ(y) ifψ(y)∈Z, and z0∈Z if not,

for some fixed elementz0∈Z without affecting the fact thatφZ =ψ◦φY.