1.11. Take n = };?Il or a separable metric space in Exercise 10 and let 9 be the class of all open sets. Let ;Y(' be a class of real-valued functions on n satisfying the following conditions.

(a) 1 E ;/(' and 1D E ,/(' for each D E ~;

(b) dr' is 'a vector space, namely: if fIE ;Yr', f 2 E ,'I(' and CI, C2 are any two real constants, then CI f 1 + c2f 2 E ,It;

2.2 PROBABILITY MEASURES AND THEIR DISTRIBUTION FUNCTIONS I 21

(c) £' is closed with respect to increasing limits of positive functions, namely: if / n E £',0 < / n < /n+l for all n, and / = limn t / n <

00, then / E :J(.

Then £' contains all Borel measurable functions on Q, namely all finite valued functions measurable with respect to the topological Borel field (= the minimal B.P. containing all open sets of Q). [HINT: let {i' = {E C Q: 1£ E df'};

apply Exercise 10 to show that -e contains the B.P. just defined. Each positive Borel measurable function is the limit of an increasing sequence of simple

(finitely-valued) functions.]