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2). One way round the difficulty is simply to take the "completion" of «5'([0, 1]) in I . 111' This procedure certainly yields a Banach space, but it does not provide an intuitive guide to the properties of the functions which the space will contain, and indeed it is not even obvious 37 38 2 LEBESGUE INTEGRATION AND THE 2" p SPACES whether the elements of the abstract completion will be functions at all. Another possibility is to enlarge the set offunctions to include all those which are Riemann integrable.

21 Definitions. A Banach space is said to be separable iff it contains a countable dense set, that is iff there is a set S = U ~ } : : : ~ in :YJ such that for each E: > 0 and f E fJ6 there is an f n E S with Ilf- j ~ I < E. 22 Lemma. (i) If n is a closed bounded subset of [RII, (6(n) with the Slip norm is separable. (ii) ( p is separable for 1 :( p < co. (iii) (C(J is not separable. 26 1 BANACH SPACES Proof We prove (i) and leave the others as exercises. The polynomials with rational coefficients are a countable set and are dense in the set of all polynomials.

Therefore X is measurable. Note that itfollows that for the Lebesgue o-algebra on IR, the rather unpleasant function f(x) = 1 (x irrational), f(x) = 0 (x rational) is measurable. 4 Example. Take Y' to be the Lebesgue o-algebra on IR", and let f: IR" -> IR be continuous. 9) which will be proved in 'the next chapter, f- 1((IX, 00)) is open. Therefore, as Y' contains all open sets, we conclude that a continuous function is measurable. 5 Example. Assume that X = [0, 00) and let Y' be the Lebesgue o-algebra.