By Kôsaku Yosida (auth.)
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Theorem 2 (Jordan's decomposition). Let cp E A (5, m) be real-valued. Then the positive variation V (cp; B), the negative variation V (cp; B) and the total variation V(cp; B) are count ably additive on B. Moreover, we have the Jordan decomposition cp(B) = V(cp; B) + V(cp; B) for any BE m. (11) Proof. Let {Bn} be a sequence of disjoint sets Em. I B n, we have cp (B) n=1 and hence v(cp; = n~ Bn)< n~ V(cp; Bn). 00 ~ n=1 Em 00 cp (B (\ Bn) < ~ V (cp; Bn) n=1 On the other hand, if CnE is a subset of Bn (n = 1, 2, ...
Constitutes a normed linear space (c) by the norm IIxll = sup Ix(n)i .. } such that constitutes a normed linear space (lP) by the norm C~ 1~.. lprp· As an abstract linear space, it is a linear subspace m, of C({l, 2, ... }). It is also a special case of LP(5, m) in which m({l}) = m({2}) = ... = l. (l00) = (m): As in the case of L OO (5), we shall denote by (lOO) the linear space C({1,2, ... }), normed by IIxll=suplx(n)1 =supl~"I. n (l00) is also denoted by (m). n The Space of Measures. Let mbe a a-ring of subsets of 5.
For any Lebesgue integrable function I (x) in G and e > 0, there exists a continuous function C. (x)"# o}a is a compact subset of G and f G 2* I/(x} - C. (x) I dx < e. o. Preliminaries 20 Remark. Let m be a Baire measure on a locally compact space S. A subset Z of 5 is called a set of m-measure zero if, for each E > 0, there is a Baire set B containing Z with m (B) < E. One can extend m to the class of m-measurable sets, such a set being one which differs from a Baire set by a set of m-measure zero.