PROBABILITY MEASURES ON METRIC SPACES. Volume 3 in by K. R. PARTHASARATHY

By K. R. PARTHASARATHY

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5 Conjugate Pairs of Additive Closure Operators 45 completely additive closure operators with respect to R. T h e n for all sets T C A and S C B , the following properties hold: Proof: We prove only (i7)and (ii'); the others are dual. ). 4 (v). 4 we have ~ ( L ( SC)p)( ~ ( y ~ ( S = (v). 4 (v) and our assumption. c c c ). 4 (iv'). In addition, ~ ( L , ( S= ~ ( L ( S )Altogether ). we obtain y 2 ( p ( ~ ( S ) )C) ~ ( L ( S )The ) . opposite inclusion is always true, since y2 is a closure operator.

T h e n for all sets T C A and S C B , the following properties hold: Proof: We prove only (i7)and (ii'); the others are dual. ). 4 (v). 4 we have ~ ( L ( SC)p)( ~ ( y ~ ( S = (v). 4 (v) and our assumption. c c c ). 4 (iv'). In addition, ~ ( L , ( S= ~ ( L ( S )Altogether ). we obtain y 2 ( p ( ~ ( S ) )C) ~ ( L ( S )The ) . opposite inclusion is always true, since y2 is a closure operator. Conversely, S ,u(L(S))implies y2(S) C y 2 ( p ( ~ ( S ) )=) ~ ( L ( Sby ) ) the , extensivity of p ~the , monotonicity of 7 2 and our assumption.

2, we see that V {T, I j E J) = LIP'( U Tj) = jEJ %/LJ j E J) and therefore K,I,I is closed under the supremum operation of EL,. We consider the relation {T x p ( T ) 1 T E U), Ru := U which we will prove is a Galois closed subrelation of R. First, for each non-empty T E U we have p(T) = {s E B I V t E T((t,s) E R)), so that T x p ( T ) R. Therefore Ru R. To show that the second condition of the definition of a Galois closed subrelation is met, we let (p', L') be the Galois connection between sets A and B induced by Ru, and assume that p' (T) = S and L'(S)= T for some T A and S B.

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