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**Example text**

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, ... ), then we have v(cp; n~ Bn) > = n~cp(Cn) m cpC~ Cn) and so V(cp;n~Bn» n~V(CP;Bn). Hence we have proved the countable additivity of V (cp; B) and those of V (cp; B) and of V (cp; B) may be proved similarly. To establish (11), we observe that, for every C Em with C <;" B, we have cp(C) = cp(B) -q;(B - C) < q;(B) - V(q;; B) and so V(cp; B) < q;(B) - V(q;; B).

Consider the set A (5, m) of all real- (or cornplex-) valued functions q;(B) defined on msuch that Iq; (B) I oF 00 for every BE (7) m, q; C~ Bj) = j~ q; (Bj) for any disjoint sequence {Bj} of sets Em. (8) A (5, m) will be called the space of signed (or complex) measures defined on (5, m). Lemma 2. Let q; E A (5, m) be real-valued. Then the total variation of rp on 5 defined by (9) V(q;; 5) = V(q;; 5) + I~(q;; 5)1 is finite; here the positive variation and the negative variation of q; over B E mare given respectively by V (q;; B) = sup q; (Bl) and B,);B E" (q;; B) = inf q; (B l ) .

The minimum of the function f (c) = ~ + ;, -c for c > 0 is attained only at c = 1, and the minimum value is O. By taking c = ab-1/(fJ- 1) we see that the Lemma is true. The proof of (2). We first prove Holder's inequality f Ix(s) y(s)1 < (J IX(s)IP)l/P. (J ly(s)IP')l/P' (5) (for convenience, we write f z(s) for! z(s) m(ds)). e. and so (5) would be true. B in (4) and integrating, we obtain Jlx(s) y(s) AB I < ~ AP + ~ BP' _ 1 which implies = P AP P' BP'- (5). Next, by (5), we have flx(5) + y(sW < + y(S)IP-l .