Filippov Implicit Function Theorem for Quasi-Caratheodory Functions

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1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 214, ARTICLE NO. AY Filippov Implicit Fuctio Theorem for Quasi-Caratheodory Fuctios M. Didos ˇ ad V. Toma Faculty of Mathematics ad Physics, Comeius Uiersity, Mlyska Dolia, 84215, Bratislaa, Sloak Republic Submitted by Bria S. Thomso Received March 21, 1996 Several Filippov type implicit fuctio theorems are kow for Caratheodory fuctios fž t, x., i.e., all fž, x. are measurable ad fž t,. are cotiuous. We prove some geeralisatios of this theorem supposig oly each fuctio fž t,. to be quasicotiuous with closed values Academic Press 1. INTRODUCTION The implicit fuctio theorem proved i 1959 by A. F. Filippov i 1 serves as a importat tool i the optimal cotrol theory. It assumes however some cotiuity coditios which are sometimes restrictive. I this paper we prove that the coclusios of Filippov s theorem remai true if we weake the cotiuity to quasicotiuity ad assume moreover closed values of a fuctio. The quasicotiuity aloe seems to be too weak to obtai some reasoable coclusios, because such a fuctio eed ot be eve measurable. The class of quasicotiuous fuctios with closed values cotais for example piecewise cotiuous fuctios which fit well for the optimal cotrol theory. We shall deote by T or more precisely Ž T, A. a abstract measurable space with a give -algebra A of subsets of T. Ifa-fiite measure is give o A we say that T is a -fiite measure space. We suppose X, Y to be topological spaces, that ofte will be Ž pseudo. metrizable. We call X a Polish space if it is separable ad metrizable by a complete metric. X is * address: didos@fmph.uiba.sk. address: toma@fmph.uiba.sk X97 $25.00 Copyright 1997 by Academic Press All rights of reproductio i ay form reserved.

2 476 DINDOS ˇ AND TOMA called a Sousli space if it is metrizable ad is a cotiuous image of a Polish space. X A set-valued fuctio F : T 2 4 is called a multifuctio from T to X ad we deote it by the symbol F : T X which remids us that F itermediates a correspodece of each poit of the departure set T with two or more poits of the target set X ad, simultaeously, icites us to cosider the values Ft Ž. as subsets of the space X ad ot merely as the poits of the power set 2 X, which usually iherits less structural properties tha X has. The graph Ž i T X. of the multifuctio F is the set GrŽ F. Ž t, x. T X : x Ft 4 ad whe o cofusio is possible we shall idetify the multifuctio F with GrŽ F. like may authors do Žsee, for example, 2 which is our stadard referece for multifuctios.. We say that a multifuctio F : T X is measurable Žweakly measurable,. B-measurable, C-measurable if the iverse image F Ž B. tt: Ft Ž. B4is a A-measurable subset of T for every closed Žope, borel, compact. subset B i X. If GrŽ F. A B where B is the borel -algebra o X, we say that F is graph measurable. 2. QUASICONTINUOUS FUNCTIONS AND CLOSED VALUES OF A FUNCTION The otio of quasicotiuity Žsee, e.g., the survey article. 5, turs out to be useful i our geeralizatio of Filippov s theorem. DEFINITION 1. Let X, Y be topological spaces. A fuctio f : X Y is said to be quasicotiuous at a poit x X if for every eighborhood V of the image fž x. ad every eighbourhood U of x there is a oempty ope set W U such that fw V.If f is quasicotiuous at each poit of its defiitio domai X, we say that f is a quasicotiuous fuctio. As the ope set W i the above defiitio eed ot to be a eighbourhood of the poit x it is clear that quasicotiuous fuctio eed ot to be cotiuous. For example, ay piecewise cotiuous fuctio f : Y is quasicotiuous. It is kow that quasicotiuous fuctio eed ot be eve measurable with respect to a metric measure. Usig quasicotiuity we ca defie a quasi-caratheodory fuctio. DEFINITION 2. Let T be measurable ad X, Y be topological spaces. We call a fuctio f : T X Y quasi-caratheodory if for each x X x the fuctio f : t fž t, x. is ABY-measurable ad for each t T the fuctio f : x fž t, x. t is quasicotiuous. Quasi-Caratheodory fuctios share some importat properties of Caratheodory fuctios. The properties proved i the ext two propositios will be used i the proof of Filippov type theorems.

3 QUASI-CARATHEODORY FUNCTIONS 477 PROPOSITION 1. Let X be a separable topological space, Y be ay topological space, ad f : T X Y be a quasi-caratheodory fuctio. If we chose a ope set V i Y such that for each t T, fžt4 X. Vthe the multifuctio f : T X defied by is weakly measurable. FŽ t. xx : fž t, x. V4 Proof. Let us cosider a ope set U X ad a coutable dese subset D U. First we prove that F Ž U. F Ž D.. Sice U D the iclusio F U F Ž D. is obvious. To prove the opposite iclusio we suppose that Ft Ž. U. Let us choose a poit xft Ž. U. The U, V are ope eighbourhoods of the poits x, f Ž x. t, respectively, ad owig to the quasicotiuity of ft there exists a oempty ope set W U with f Ž W. t V. Makig use of desity of the set D we have D W so there must exist a poit x D W such that Ž f x. V,so xft Ž. D. It meas Ft Ž. t Dad the iclusio F U F Ž D. is proved. To fiish the proof it suffices to esure the A-measurability of F Ž D.. We ca write F Ž D. t T x D : x FŽ t. 4 4 x t xd tt : f x V f V xd ad the last set is measurable as a coutable uio of measurable sets. PROPOSITION 2. Let Ž T, A. be a measurable space, X be a topological space, ad Ž Y, d. be a separable pseudometric space. If g : T Y is a measurable fuctio ad f : T X Y is a quasi-caratheodory fuctio the the fuctio is also a quasi-caratheodory fuctio. hž t, x. d gž t., fž t, x. Proof. Let BŽ X. be the Borel -algebra o X. The the projectio mappig : T X T is measurable with respect to the product -algebra A BŽ X.. The composed mappig g : T X Y is quasi- Caratheodory Ževe Caratheodory, because it is costat o each sectio 4 t X.. So the mappig Ž g, f. : T X Y Y Ž t, x. gž Ž t, x.., fž t, x.

4 478 DINDOS ˇ AND TOMA is quasicotiuous i x. The measurability i the variable t is a immediate cosequece of the equality BŽ Y Y. BŽ Y. BŽ Y., which is true because of the coutable base of the topology i Y. The metric d : Y Y is a cotiuous mappig, therefore the compositio dž g, f. is a quasi-caratheodory fuctio. The quasicotiuous fuctios are rather geeral ad we have to seek for some restrictios. We have iveted the otio of a closed value of a fuctio which turs out to be useful ad, whe combied with the quasicotiuity, it allows us to prove a geeralized Filippov theorem. DEFINITION 3. Let X, Y be first coutable topological spaces. We say that a poit y Y is a closed alue of a fuctio f : X Y if for each sequece Ž x. the followig implicatio holds: X Y x x ad fž x. y implies fž x. y. Every poit y Y Im f is a closed value of the fuctio f so we could have limited the choice of y i the previous defiitio to the closure of the rage of the fuctio f, i.e., y Im f. The otio of a closed value is a kid of localizatio of the closed graph of a fuctio because it meas that GrŽ f. X y4 GrŽ f. X y 4. The followig propositio is straightforward ad the proof is left to the reader. PROPOSITION 3. Let X, Y be first coutable topological spaces. Eery poit y Y is a closed alue of the fuctio f : X Y iff the graph of f is a closed set. As a closed value of a fuctio f eed ot be at all the value of f, maybe it would be more appropriate to say that the graph of f is closed at the poit y. Remark 1. If a fuctio f is cotiuous the each f x is a closed value of f. The ext example shows that quasicotiuous fuctios with closed values eed ot be cotiuous. EXAMPLE. The fuctio f : defied by if x 0 fž x. x 0 if x0 is quasicotiuous with 0 as a closed value but f is ot cotiuous.

5 QUASI-CARATHEODORY FUNCTIONS 479 The ext lemma is the core i the proof of the geeralized Filippov theorem. LEMMA 1. Let X be a first coutable topological space ad Ž Y, d. be a pseudometric space. Let y Y be a closed alue of a fuctio f : X Y. If we deote by G the sets the we hae 5 1 G x X : d Ž fž x., y. ½ N G G. Proof. The iclusio is obvious. To prove the opposite iclusio let x G.So, xg ad because of first coutabil- X ity of X there is a sequece x G such that x x. Owig to Ž. Y xg we have, d f x, y ad therefore f x y. Usig the fact that y is a closed value the equality fž x. y holds true. So we have proved that 1, x G ad hece x G. Now we have prepared to prove the mai theorem of this paper. THEOREM 1. Let X be separable metric space ad Ž Y, d. be a pseudometric space. Let f : T X Y be a quasi-caratheodory fuctio, : T Xbe a compact-alued measurable multifuctio, ad g : T Y a measurable fuctio such that g Ž t. is a closed alue of the fuctio ftž x. fž t, x. ad gž t. f Ž t. tt. t The exists a measurable selector : T X of the multifuctio such that gž t. fž t,ž t.. tt. Proof. Every measurable selector : T X of the multifuctio HŽ t. Ž t. xx : fž t,w. gž t. 4 tt would satisfy the coclusio of the theorem. We prove the existece of such a selector provig that H verifies the hypothesis of the Kuratowski ad Ryll-Nardzewski selector theorem. Sice gž. t is a closed value of f, t

6 480 DINDOS ˇ AND TOMA the multifuctio 4 Ž t. G t xx : f t, x g t f g t is a closed-valued multifuctio. To prove the measurability of G let us cosider the multifuctios G : T X defied by 1 GŽ t. ½xX : dž fž t, x., gž t.. 5. The fuctio ht,x džfž t, x., gt. is a quasi-caratheodory fuctio followig Propositio 2, so the multifuctios t GŽ t. are weakly mea- surable. Owig to Lemma 1 we have GŽ t. G Ž t. G Ž t.. Therefore the multifuctio H : T X, HŽ. t Ž. t Gt Ž. t Ž. GŽ t. is measurable as the coutable itersectio of cout- ably may compact-valued measurable multifuctios. Remark 2. If we do ot suppose that gt Ž. fžt Ž.. t for all t T we ca defie T t T : gt Ž. fžt Ž..4 0 t ad we ca claim the existece of the measurable fuctio : T X such that gt Ž. fžt,ž.. 0 t tt 0. If we suppose more about the spaces T or X, the multifuctio eed ot have compact values. A easy cosequece of Theorem 1 is the followig COROLLARY 1. If X is a -compact space the we ca require the multifuctio i Theorem 1 to be oly closed-alued. Repeatig the proof of Theorem 1 ad usig the selector theorem of Himmelberg 2, Theorem 5.7 istead of Kuratowski ad Ryll-Nardzewski, we ca eve drop the assumptio o closed values. THEOREM 2. Let T be a -fiite measure space, X be a Sousli space, ad Y a separable metric space. If the fuctios f, g are such as i Theorem 1 ad the multifuctio : T X is graph measurable the there is a measurable fuctio : T X such that Ž. t Ž. t ad gž. t fžt,ž.. t for almost all t T. If the fuctio f depeds oly o oe variable x ad we assume the cotiuum hypothesis, we obtai some liftig theorem which geeralizes those proved i 2 uder stroger assumptios. COROLLARY 2. Let X be a separable metric space ad Y a Hausdorff Space. If f : X Y is a quasicotiuous fuctio, : T X is a closed-alue

7 QUASI-CARATHEODORY FUNCTIONS 481 C-measurable multifuctio, ad g : T YisaC-measurable fuctio with closed alue gž. t fžž.. t of the fuctio f for eery t T the there exists a C-measurable selector : T X such that gž. t fž Ž.. t for all t T. REFERENCES 1. A. F. Filippov, O ekatorych voprosach teorii optimalovo regulirovaia, Vestik Mosko. Ui. 2 Ž 1959., C. J. Himmelberg, Measurable relatios, Fud. Math. 87 Ž 1975., A. Kucia ad A. Nowak, O Filippov type theorem ad measurable iverses of radom operators, Bull. Polish Acad. Sci. Math. 38, No. 12 Ž 1990., K. Kuratowski ad C. Ryll-Nardzewski, A geeral theorem o selectors, Bull. Acad. Polo. Sci. Ser. Sci. Math. Astroom. Phys. 13 Ž 1965., T. Neubru, Quasi-cotiuity, Real Aal. Exchage 14 Ž ,