ABSTRACT
Then ( i ) , ( i i ) and ( i i i ) aTe equivalent, and either of them implies ( i u ) . Furthermore, ( i v ) implies ( i ) if Q is universally complete. PROOF In order t o see that (i) implies ( i i ) let U C X be open, and let D C X be dense and separable. For every x E D n U choose a, with d ( x , U C ) / 2 < a, < d ( x , U c ) , say. T h e n B(x,cr,) c U and U = UZED,, B ( x , a , ) . T h u s {W : C ( W ) n u # 0 ) = u,,,{w : C ( W ) n B ( X , Q , ) # 0 ) = u z t D { w : d ( x , C ( w ) ) < a,), which is measurable. T o show that ( i i ) iniplies ( i ) let x E X and 6 > 0. T h e n {w : d ( x , C ( w ) ) < 6 ) = { w : C ( w ) n B(x , 6 ) # 0) E Q b y ( i i ) applied wi th U = B ( x , 6 ) . T h i s being true for every 6 > 0, w tt d ( x , C ( w ) ) is measurable, which is ( i ) . T o prove t he equivalence o f ( i ) and ( i i i ) , first suppose that ( i ) holds. T h e n
p a p h ( ~ 9 = { ( x , w ) : x E c 6 ( w ) } = { ( x , w ) : d ( x , C ( w ) ) < 61,
hence measurability o f graph(C6) follows from joint measurability o f ( x , w ) H d ( x , C ( w ) ) , see Renlark 2.2. I f o n t he other hand is a measurable subset o f X x R for every 6 > 0 , t hen for every x E X it holds that { w : d ( x , C ( w ) ) < 6 ) E 9 for all 6 > 0. But this means measurability o f w tt d ( x , C ( w ) ) , which is ( i ) . T o see tha t ( i ) implies ( i v ) note that graph(C) = { ( x , w ) : x E C ( w ) ) = { ( x , W ) : d ( x , C ( w ) ) = 0 ) . For the proof that ( i v ) implies ( i ) in case 9 is urliversally corrlplete we need the Projection Theorem (Theorem 2.12).
