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9. 5 would show that lim fn = 1 weakly in C(X), and therefore that 1 = I(1) would equal lim I(fn), which is 0. This contradiction forces the conclusion that X is non-compact. Finally assume that X is non-compact, c. r. , and expressible as the union of a sequence (Wn) of relatively compact open sets. It may be assumed that the Wn increase with n. Moreover, by dropping terms of the sequence (Wn), if necessary, it may be arranged that Wn is a proper subset of Wn+l. Choose arbitrarily xn from Wn+1\Wri On the other hand, since X is c.

1. 7. 11. Exercise. Let (f n) be a sequence of real-valued measurable functions on X, and let E be the set of points x E X for which the sequence (fn(x)) is convergent in R. Show that E is measurable. Suppose that f: X -R is measurable and that g: R - R is continuous. Prove that gof is measurable. 1. 7. 12. Exercise. Remark. A little surprisingly; perhaps, fog need not be measurable, see Halmos [1], p. 83. 28 1. 13. Exercise. ao, =0. n- oo am,n Select any strictly increasing sequence k 1- nk of positive integers such that 00 k 00 (k2ank' nk+1)p Let (fn)nEN be any sequence of elements of p(fm - fn) p such that am,n for every m, n E N.

3. Daniell's work was largely neglected for some twenty years, at the end of which period interest was revived by Bourbaki (c. 1937), presumably when plans were being laid for the relevant sections of his future work [1] in this field. Publication of this work began in 1952, since when further volumes have appeared. Between 1937 and 1952, some of the ideas filtered through to the mathematical world at large (in the guise mainly of books and research papers authored by individual members of the Bourbaki group, such as Weil and H.