ZESZYTY NAUKOWE WYŻSZEJ SZKOŁY PEDAGOGICZNEJ w BYDGOSZCZY Problemy Matematyczne 1985 z. 7
WŁODZIMIERZ ŚLĘZAK WSP w Bydgoszczy
SOME REMARKS ABOUT CEDER-LEVI THEOREM
Let X and Y be topological spaces. Recall that a multi function F: X — ^ Y is said to be of lower class oC if F~(U):= £x: F(x)ft U = 0 J is a Borel set in X of additive class o c for each open set U in Y. Interrelations between above notion and usual Baire classification were investigated in [ 3 ] .
flJ (»ее also [2J ) is inductively proved, that the multi function of lower class oC with convex but not necessarily closed values in finite dimensional linear space possess© a Borel oC selector. We may ask whether the range space in Ceder-Levi theorem can be generalized. Some negative results in this direction were given in [в] . This note is devoted to positive one. Namely, we exhibite this fact, that the Michael’s [6J methods gives more powerful theorem on Borel
oC selectors.
Let Y be a Banach space or, more generally a Frechet space. If К is a close, convex subset of Y, then a supporting set of К is, by definition, a closed, convex proper subset S of К (_in particular a singleton), such that if an interior point of a segment in К belongs to S, then the whole segment is contained in S. The set I(K) of all elements of К which are not in any supporting set of К will be called the inside of K. The family
D(Y) = ^B CY : В = conv В and В =? I (Cl B>] is seemingly the adequate range space for the Ceder-Levi theorem.
Emphasize that every convex set which is either closed, or has an interior point,or is finite deimensional, belongs to D (Y ) (see C6J, p. 372,) . We prove the following
improve-151
ment of the Ceder-Levi theorem:
THEOREM. Let F:X — D(Y) where X la a perfectly normal topological apaoe and Y la a aeparahle Frechet apace.
If F la of lower claaa of > 0 , then F haa a Borel aelector. Proof. Define F:X Y by formula F <x) = Cl F(x) ; what we ■mat find la a Borel of aelector f :X — * Y auch that
f (х)61(£ (x>) for every i & l , Obvioualy F la alao of lower claaa of. Thua, In virtue of the theorem 4 of L5j , F haa the ao-oalled Caataing's representation. Namely, there exist functiona f ^ : X - > Y , 1*1,2,..., auch that eaoh f^ la a Borel of function and we have the equality
F(x)= Cl ({f± ():i=1,2,...); on the whole X.
Now, let g ± ( x ) = +
fA (x)- f ^ x )
max(l, d (f ^ (x ) , f^x)))
i = 9 where d denote the invarlcmt metric on Y« Put f (x) =
27
2"1 g (X) .i=1 1
An lnapection of the proof of the lemma 5.1 in féj ahowa that f(x)4l (f(x))c F(x) for every xe_X. Since the series defining f :X —^ Y converges almost uniformly on X, it follows that f is also of Borel class of * and thus has all the required properties.
REFERENCES
[1] Ceder J.and Levi S., On the search for Borel 1 selections, (to appear)
[2] Ceder J, Some problems on Baire i selections, Real Analysis Exchange, vol. 8.2 (19 8 2 -8 3 ), p.502-503
[3 ] Kuratowski K., Some remarks on the relation of classical set-valued mappings to the Baire clasaification, Colloqu ium Math. XLII C1979), p.274-277
[4] MAgerl G., A unified approach to measurable and continuous aelections, Transaction AMS, vol. 245 (1978), p.443-452 [5 ] Maitra A. and Rao B.V., Generalizations of Castaing's
152
[6 ] Michael В., Continuous selections I, Annals of Math., vol. 6 3 , 2 [19 5 6 ), p.3 6 1 -3 8 2
[7} Michael E.and Pixley C., A unified theorem on continuous selections, Pacific J. of Math., vol.87 , 1 0 980), p. 187-
- 1 8 8
[8] Ślęzak W., Some counter examples in Multifunction Theory, Real Analysis Exchange, vol. 8 , 2 (1982-83), p.494-501
