Isaac V. Shragin
On some theorem on measurable selection
Abstract. A new proof of some results of V.L. Levin on measurability of projection and on measurable selection is given.
2000 Mathematics Subject Classification: 28A05, 28A20.
Key words and phrases: A -operation, analytic set, mesurable selection, Polish space, Souslin space.
1. Introduction. In [1] (see also [2]) V.L. Levin obtained some significant results on measurability of projection and on measurable selection. Here we suggest other method of the proof of Theorem 2 and Corollary 2 from [1]. This method is based on arguments from [3] (Sections 8.4 and 8.5).
2. Preliminaries. Remind the notion of A -operation [4]. Denote by N the set of all infinite sequences n = (n1, n2, . . .) of positive integers, i.e. N = NN. The A -operation (or Souslin operation) applied to countable family of sets
nAn|k: n|k = (n1, . . . , nk) ∈ Nk, k = 1.2, . . .o
produces the set
[
n∈N
\∞ k=1
An|k.
If E is some collection of sets, then A (E ) denotes the family of all sets obtained with the help of A -operation over sets from E . Evidently, E ⊂ A (E ), and it is known that A (A (E )) = A (E ).
We shall apply the A -operation to sets from Polish space, i.e. separable topo- logical space that can be metrized by means of a complete metric. It is known ([3],
Prop. 8.1.3) that topological product Z × X of Polish spaces Z and X is Polish too. Note also that each closed or open subspace of a Polish space is Polish ([3], Prop. 8.1.1).
The important examples of Polish space are the Baire space N = NN and Cantor space C = {0; 1}N equipped with countable products of discrete topologies in N and {0; 1} respectively. It is convenient to consider N and C with metric d(m, n) = (min{k : mk 6= nk})−1 where m = (m1, m2, . . . ), n = (n1, n2, . . . ), m6= n.
A set A in Polish space X is called analytic (or Souslin) if A is continous image of a Polish space Z , i.e. A = f(Z ), where f : Z →X is continuous. But every Polish space is continuous image of the Baire space N ([3], Prop. 8.2.7). Therefore it is possible to define an analytic set in Polish space X as a continuous image of N . Note that B(X )⊂ S (X ) ([3], Prop. 8.2.3), where B(X ) and S (X ) are the families of all Borel and Souslin sets in Polish space X , respectively.
It is well known that S (X ) = A (F (X )), where F (X ) is the family of all closed sets in the Polish space X (([5], Theorem 25.7, see also [4], Ch.13,§1, Theorem 4)). But since B(X ) ⊂ S (X ) and A (A (F(X ))) = A (F(X )), then S (X ) = A (B(X )).
If E is some collection of subsets of a set E, we denote by σ(E ) the σ-algebra on E generated by E . If (Ei, Σi), i = 1, 2 are measurable spaces, we call a function f : A→E2 (with A ⊂ E1) (Σ1, Σ2)-measurable if (∀ B ∈ Σ2) f−1(B) ∈ Σ1. We denote by < Σ1, Σ2> the collection of all (Σ1, Σ2)-measurable functions. By Σ1⊗Σ2 we denote the product of σ-algebras Σ1 and Σ2, i.e. the σ-algebra on E1 × E2 generated by the family {A × B : A ∈ Σ1, B∈ Σ2}.
3. The main results. Futher we shall use the following theorem ([3], Theorem 8.5.3)
Theorem 3.1 Let Z and X be Polish spaces, D∈ S (Z × X ), D0= prZ D (pro- jction of D on Z ). Then there exists a function ψ : D0→X such that grψ ⊂ D and ψ ∈< σ(S (Z )), B(X ) >. Here grψ denotes the graph of ψ, i.e. {(z, ψ(z)) : z∈ D0}.
The following lemma is a modification of Lemma 8.4.5 from [3].
Lemma 3.2 Let (T, Σ) be a measurable space, X be a Polish space, C ⊂ T × X with C ∈ A (Σ ⊗ B(X )). Then there exist a function h : T →C = {0; 1}N and a set D ∈ S (C × X ) such that h ∈< Σ, B(C ) > and C = H−1(D), where H = h× idX : T × X →C × X , i.e. H(t, x) = (h(t), x).
Proof Since C ∈ A (Σ⊗B(X )), then to each n|k = (n1, . . . , nk) ∈ Nk, k = 1, 2, . . . corresponds a set Cn|k∈ Σ ⊗ B(X ) such that
C = [
n∈N
\∞ k=1
Cn|k.
As Cn|k∈ Σ⊗B(X ) then ([6], Statement 13.9) for every Cn|kthere exists a sequence of ”rectangles” Cn(p)|k = A(p)n|k× Bn(p)|k, p = 1, 2, . . . , with A(p)n|k ∈ Σ, Bn(p)|k ∈ B(X ), such that Cn|k ∈ σ{Cn(p)|k: p = 1, 2, . . . }.
Arrange all sets Cn(p)|k, n|k ∈ Nk, k, p ∈ N, in a sequence {Qm= Am× Bm: m = 1, 2, . . . }. Then ∀(n|k)Cn|k ∈ σ{Qm: m = 1, 2, . . . }, so that C ∈ A (σ{Qm : m ∈ N}).Define the function h : T →C by putting
h(t) = (χA1(t), χA2(t), . . . ),
where χA denotes, as usually, the characteristic function of the set A ⊂ T . Put further
Em= {(z1, z2, . . . )∈ C : zm= 1}, m = 1, 2, . . . .
Then h−1(Em) = Am∈ Σ, m = 1, 2, . . . . It is not difficult to show that the ”clopen”
(i.e. closed and open) sets Em generate the σ-algebra B(C ), i.e. B(C ) = σ{Em: m∈ N}. Hence h ∈< Σ, B(C ) >. From here it follows taht H ∈< Σ⊗B(X ), B(C × X ) > (it is necessary to take in account that B(C × X ) = B(C ) ⊗ B(X ), [3], Prop. 8.1.5).
Put F = {H−1(F ) : F ∈ B(C × X )}. Then F is a σ − algebra on T × X , and F ⊂ Σ ⊗ B(X ). In addition, since
Qm= h−1(Em) × Bm= H−1(Em× Bm) ∈ F, m = 1, 2, . . . ,
then σ{Qm: m ∈ N} ⊂ F, whence C ∈ A (F). It means that there exists a family of sets Dn|k∈ B(C × X ), n|k ∈ Nk, k = 1, 2, . . . , such that
C = [
n∈N
\∞ k=1
H−1(Dn|k) = H−1
[
n∈N
\∞ k=1
Dn|k
.
Put D = S
n∈N
∞T
k=1
Dn|k. Then D ∈ A (B(C × X )) = S (C × X ) and C =
H−1(D).
Theorem 3.3 Let (T, Σ) be a measurable space, X be a Polish space, C ⊂ T × X with C ∈ A (Σ ⊗ B(X )). Then C0 = prTC ∈ A (Σ), and there exists a function ϕ : C0→X such that grϕ ⊂ C and ϕ ∈< σ(A (Σ)), B(X ) >.
Proof By Lemma 3.2 there exist a function h : T→C = {0; 1}N and a set D ∈ S (C×X ) such that h ∈< Σ, B(C ) >, and C = H−1(D), where H(t, x) = (h(t), x).
In what follows we denote by h−1(E ) the family {h−1(E) : E ∈ E }, where E is a collection of some subsets of C . So that we have
(1) h−1(S (C )) = h−1[A (B(C ))] = A [h−1(B(C ))] ⊂ A (Σ).
The projection of D on C is an analytic set (as continuous image of analytic set), i.e. D0= prC D ∈ S (C ). From here and (1) it follows that h−1(D0) ∈ A (Σ). But
h−1(D0) = {t ∈ T : (∃x ∈ X )(h(t), x) ∈ D} = prTH−1(D) = C0.
Thus C0 = h−1(D0) and C0 ∈ A (Σ). By Theorem 3.1 there exists a function ψ : D0→X such that grψ⊂ D, and ψ ∈< σ(S (C )), B(X ) >. Put ϕ(t) = ψ(h(t)), t∈ C0(so that ϕ : C0→X ) and show that ϕ ∈< σ(A (Σ)), B(X ) >.
Let B ∈ B(X ). Then ϕ−1(B) = h−1(ψ−1(B)). Since ψ−1(B) ∈ σ(S (C )), then ϕ−1(B) ∈ h−1[σ(S (C ))] = σ[h−1(S (C ))] ⊂ σ(A (Σ)), because of (1), i.e.
ϕ∈< σ(A (Σ)), B(X ) >.
Finally show that grϕ ⊂ C. Let (t, x) ∈ grϕ, i.e. t ∈ C0, x = ϕ(t) = ψ(h(t)).
Then h(t) ∈ D0, so that (h(t), ψ(h(t))) ∈ grψ ⊂ D, i.e. H(t, ϕ(t)) ∈ D. Hence
(t, ϕ(t)) ∈ H−1(D) = C.
Now we shall generalize Theorem 3.3 to situation when X is a Souslin space, i.e.
X is a topological space which is a continuous image of a Polish space (note that X is not necessarily Hausdorff space).
Theorem 3.4 The conclusion of Theorem 3.3 remains valid if the space X is Souslin.
Proof There exist a Polish space P and a continuous map q : P→X such that X = q(P ). Define the map Q : T × P →T × X with Q(t, p) = (t, q(p)). If A ∈ Σ, B ∈ B(X ), then
Q−1(A × B) = A × q−1(B) ∈ Σ ⊗ B(P ),
i.e. Q ∈< Σ ⊗ B(P ), Σ ⊗ B(X ) >. Hence Q−1(C) ∈ A (Σ ⊗ B(P )). By Theorem 3.3 prTQ−1(C) ∈ A (Σ). But it is easy to check that prTQ−1(C) = prTC. Hence prTC ∈ A (Σ). In addition, by Theorem 3.3 there exists a map Π : prTC→P such that grΠ ⊂ Q−1(C) (i.e. Q(grΠ) ⊂ C) and Π ∈< σ(A (Σ)), B(P ) >.
Put ϕ = q ◦ Π : prTC→X . Then ∀ B ∈ B(X )
ϕ−1(B) = Π−1(q−1(B)) ∈ σ(A (Σ)).
So that ϕ ∈< σ(A (Σ)), B(X ) >. Finally, grϕ ⊂ C since ∀t ∈ prTC (t, ϕ(t)) =
(t, q(Π(t))) = Q(t, Π(t)) ∈ C.
Remark 3.5 Theorems 3.3 and 3.4 are especially convenient for applications (see, for example, [7], Section 7.2) if the σ-algebra Σ is closed under A -operation, i.e.
A (Σ) = Σ, since in this case σ(A (Σ)) = Σ. For example, a σ-algebra on which a σ-finite complete measure is defined, is closed under A -operation ([5], Section 29C).
References
[1] V.L. Levin, Measurable cross-sections of multivalued mappings, and projections of measurable sets, Funct. Anal. i Prilozh. 12(2) (1978), 40-45 (in Russian) [English translation: Funct. Anal.
Appl. 12(2) (1978), 108-112].
[2] V.L. Levin, Measurable cross-sections of multivalued mappings into topological spaces ,and upper envelopes of Carath´eodory integrands, Soviet Acad. Sci. Dokl. 252(3) (1980), 535-539 (in Russian) [English translation: Soviet Mathematics, Doklady, 21(3) (1980), 771-775].
[3] D.L. Cohn, Measure theory, Birkh¨auser, Boston 1993.
[4] K. Kuratowski and A. Mostowski, Set theory, North Holland, Amsterdam - New York - Oxford 1976.
[5] A.S. Kechris, Classical descriptive set theory, Springer Verlag, New York 1995.
[6] K.R. Parthasarathy, Introduction to probability and measure, Springer Verlag, New York 1978.
[7] I. Shragin, Superpositional measurability and superposition operator, Astroprint, Odessa 2007.
Isaac V. Shragin K¨oln, Germany
E-mail: marina spektor@mail.ru
(Received: 13.06.2007)
