Kucia, Anna

Nowak, Andrzej

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae UniversitatisCarolinaeAnna Kucia; Andrzej NowakOn Baire approximations of normal integrandsCommentationes Mathematicae Universitatis Carolinae, Vol. 30 (1989), No. 2,373--376Persistent URL: http://dml.cz/dmlcz/106754Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1989Institute of Mathematics of the Academy of Sciences of the Czech Republicprovides access to digitized documents strictly for personal use. Each copyof any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronicdelivery and stamped with digital signature within theproject DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czComment.Math.Univ.Carolinae 30,2(1989)373-376 373 On Baire approximations of normal integrands ANNA KUCIA, ANDRZEJ NOWAK Abstract. Let D C T x X, where T is a measurable space and X a metric space. We give conditions on T, Xjand D such that every upper semicontinuous in x and meausarable function / : D —• R is the limit of a decreasing sequence of measurable functions which are continuous in x. Keywords: Normal integrand, Baire approximation Classification: 54C30, 28A20 1. Problem. The well known theorem of Baire states that every upper semicontin-uous function on a metric space is the limit of a decreasing sequence of continuous functions. In this note we prove an analogue of this theorem for an upper semicon-tinuous functions which depends measurably on the parameter. Throughout the paper (T, T) is a measurable space, (X, d) a metric space, and D a nonempty subset of T x X. The set D is always considered with the trace cr-field {D H A\A € T ® B(X)}, where B(X) stands for the Borel a-field on X. By Dt we denote t-sections of D, i.e. Dt = {x|(t,x) € D},t € T. We shall deal with the following classes of extended real-valued functions on D: Fi(D) = {/ : D —• R\f is measurable and for each t £ T,f(t, •) is upper semicon-tinuous on Dt}, F2(D) = {/ : D —• R\ there is a sequence of measurable functions fn:D~+R such that / n + i < / n , / n ( ^ * ) is continuous on Dt and lim / n ( t , x ) = / ( t , x ) , ( t ,x) € n—->oo D). Functions from Fi(D) are called normal integrands. They are extensively stud-ied in mathematical programming. The class F2(D) also appears in optimization, especially, in stochastic dynamic programming ([7]-[9]). It is obvious that F2(D) C Fi(D) . We give sufficient conditions for the equality Fi(D) = F2(D), and point out its application. Remark. If X is separable and g : T x X —•JKisa Caratheodory map (i.e. measurable in t and continuous in a?), then g is measurable. Hence, in the definition of ^ ( T x X ) it suffices to require the measurability of / n ( - , x) instead of the product-measurability of / n . 2, Auxiliary results. In this section we quote some set-theoretical and topolog-ical results which will be used in the proof of the main theorem. Let % be a family of sets. By S(K) we denote the family of all sets obtained from % by the Souslin operation. If S(1l) = ft, we say % is a Souslin family (cf. [2], Chap.2; [12], §2). 374 A.Kucia, A.Nowak The <T-field T on T is a Souslin family provided one of the following conditions holds: (i) T is complete with respect to a a-finite measure, (ii) T is the family of all ^-measurable sets, where p is an outer measure on T, (iii) T is the a-field of universally measurable sets for some cr-field on T, (iv) T is a topological space, and T is the family of all sets with the Baire property. A metric space X is called Souslin if it is a continuous image of a Polish space. In such a space the family of all Souslin subsets coincides with S(B(X)) and S(.F), where T is the set of all closed sets in X. For a subset Z C T x X we denote by pr^Z its projection on T, i.e. PTTZ = {t|(*,a:) 6 Z for some x 6 X}. The following projection theorem is very useful. Theorem 2 . 1 . (cf. [2], Th.1.8). Suppose T is a Souslin family and X is a Souslin space. Then prTZ G T for every Z € S(T ® B(X)). We shall also use the following results. Theorem 2.2. ([4]). Let T and X be Polish spaces, A a Souslin subset ofTxX, and B a subset of A such that B is Borel in A, and Bt is open in Atlt £ T. Then B = Af)\J CnxGn, where Cn is Borel in T and Gn is open in X, n € N. Theorem 2.3. ([5], Th.2.8; [6], Th.5). Let X with the topology Q be a separable and metrizable space. Then for any sequence {Bn} of Borel sets in X there is a separable and metrizable topology Q1 on X such that Q CQ1, each Bn € Q', and the a-fields generated by Q and Q1 are the same. 3 . Main theorem. The following theorem is the main result of the paper. Theorem 3.1 . Let X be a metrizable space and D € S(T®B(X)). Then FX(D) = F2(D), provided one of the conditions holds: (i) T is a Souslin family and X is a Souslin space, (ii) T and X are Polish spaces, and T = B(T). Moreover, for each f £ Fi(D) there is g € F2(T x X) such thai g\o ~ / • PROOF : It suffices to prove that each / € Fi(D) can be extended to a function from F2(T x X). Since there is an increasing homeomorphism of R and [—1,1], we may assume |/(t,ar)| < l , ( t ,£) € D. We start with the proof under the assumption (i). By Theorem 2.1, the set P = PTTD belongs to T. Similarly, as in the proof of the theorem of Baire (see e.g. [1], p.390), we define the functions fn: PxX —• ( -co , 1] and gn : P x X -> ( -1 ,1] by the formulae /„(*,*) = sup{/(t ,y) -nd(x,y)\y € Dt}, gn(t,x) = max{/n(t,aO, - 1 -I- - } , n € N. On Baire approximations of normal integrands 375 These functions are measurable in t. Indeed, for any x £ X and r € R we have {t\fn(t, x) > r} = {t\f(t, x) — n d(xy y) > r for some y £ Dt} = = prr{(t, y) £ D\f(t, y)-n d(xy y) > r} £ T, because of the measurability of / and the projection theorem. Being decreasing, the sequence {gn} is convergent to a function g : P x X —• [—1,1], FVom the proof of the theorem of Baire we know that gn(ty •) are continuous, and g\o = / . Since gn is a Caratheodory map on P x X, it is measurable. Hence, g £ F2(P X X). It is clear that g has an extension to a function from F2(T X X). For the second part of the proof we assume that T and X are Polish spaces. Since / G Ki(D), for each rational r £ ( -1,1] the set Ar = / " ^ ( [ - V ) ) is Borel in D, and its t-section is open in Dtl t £ T. In virtue of Theorem 2.2, A' = Dn [jCnxGn, n€/V where Cn £ B(T) and Gn is open in K, n € N. Denote by Q the Polish topology of T. By Theorem 2.3, there is a stronger separable and metrizable topology Q* on T, such that Cn £ Q' for each n £ N and each rational r € (—1,1], and the a-fields generated by (/ and Q' are the same. Now we regard T x X with the new product topology, where T is equipped with Q'. In this topology each Ar is open in D and, consequently, / is upper semicontinuous. We refer to the proof of the theorem of Baire again. Let fn:T x X —> (~oo, 1] and g : T x X —> (—1,1] be defined as fn(a) = sup{/(6) - np(a,b)\b £ £>}, gn(a) = max{/„(a), - 1 + - } , n £ N, n where /o is a metric of T x K. The decreasing sequence of continuous functions {#n} is convergent to a function g : T x X —» [—1,1] such that g\o = / . Since the ^--fields generated by Q and £ ' are the same and the topology of X does not change, each gn is Borel-measurable with respect to the original topology of T x K, and continuous in x. Thus g £ F2(T x K), which completes the proof. • Remarks. 1. Related problems were studied by Dynkin ([3], Lemma 3.2), Schal ([8], §11; [10]) and &lezak ([11], Lemma 1). Our theorem generalizes corresponding results from [10] and [11]. 2. It would be interesting to know if the inclusion F\(T x X) C F2(T x X) holds for an arbitrary <r-field T and a Polish space X. 4. Application. The assumption that a function / belongs to the class F2(D) appears in some theorems in optimization ([7]-[9]). Using Theorem 3.1, we can replace this assumption by an easier verified condition / € F\(D). As an example we give one of such results. 376 A.Kucia, A.Nowak Let / : T x X —• R be measurable and bounded from above in t, and q : T x X —• [0,1] a transition probability from X to T, i.e. <?(•,£) is a probability measure on T for each x € X, and #(A, •) is measurable for each A € T . Then the function Ф ) = / /(*> *M*> *), * € X is well defined. In stochastic optimization it is interesting to know under which assumptions v is upper semicontinuous. We say that the transition probability q is 5-continuous, if for each bounded and measurable u : T —> R the function x —• / u(t)q(dt, x), x € X JT is continuous. The following result is an immediate consequence of Prop. 14.1 of Schal [8] and our Theorem 3.1. Theorem 4.1 . Suppose either (i) T is a Souslin family and X is a Souslin space, or (ii) T and X are Souslin subsets of Polish spaces and T = B(T). If f belongs to F\(T X X) and is bounded from above, and q is s-continuous, then v is upper semicontinuous. REFERENCES [1] Ash R.B., "Real Analysis and Probability," Academic Press, New York, 1972. [2] Christensen J.P.R., "Topology and Borel Structure," North Holland, Amsterdam, 1974. [3] Dynkin E.B., Stochastic concave dynamic programming, Mat.Sb. 87 (1972), 490-503; English transl.: Math.USSR-Sb. 16 (1972), 501-515. [4] Kucia A., Carathiodory type selectors, submitted. [5] Miller D.E., Borel selectors for separated quotients, Pacific J .Math. 91 (1980), 187-198. [6] Sarbadhikari H., Sri vast ava S.M., Random Tietze and Dugundji extension theorems, J. Math. Anal.Appl. (to appear). [7] Schal M., A selection theorem for optimization problems, Arch.Math. 25 (1974), 219-224. [8] Schal M., Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal, Z.Wahrsch.Verw.Gebiete 32 (1975), 179-196. [9] Schal M., On dynamic programming: compactness of the space of policies, Stochastic Pro-cess. A ppl. 3 (1975), 345-364. [10] Schal M., "Addendum to [7], [8] and [9]," Technical Report, Univ.Bonn, 1977. [11] Sle,zak W., On Carathiodory'a selectors for multifunctions with values in S-contractible spaces, Problemy Math. 7 (1986), 21-34. [12] Wagner D.H., Survey of measurable selection theorems, SIAM J .Control 15 (1977), 859-903. Silesian University, Institute of Mathematics, 40-007 Katowice, Poland (Received April 7,1989) 

On Baire approximations of normal integrands

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On Baire approximations of normal integrands

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Institute of Mathematics AS CR, v. v. i.

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