u strongly in L9, see Sec. 3. 1 + t2 is strictly convex, for k = 1, 2, ..

General Measure Theory 40 Theorem 5. Let p be a Radon measure over X and let fi, f2,... be functions in LP (X; p). (i) Suppose p > 1. ) and a subsequence {fk;} such that fk, - f weakly in LP(X; p). (ii) Suppose p = 1. Then f weakly in L'(X;lc) iff fk -1 f as measures and the a) We have fk fk's are equi-integrable in L'(X;{i) b) If SUN II fk IIL'(x;,t) < oo and the fk's are equi-integrable, then there exist f c Li (X, p) and a subsequence { fk; } such that f k, f weakly in Li(X; µ). While bounded sequences { fk } in Li (X; A) converge as measures to a mea- sure modulo passing to subsequences, they in general do not converge, even passing to subsequences, weakly in Li (X; µ), compare Sec.