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a-algebra absolutely continuous admissable concave function admissable convex function Apply lemma Apply theorem bounded function Cauchy Cauchy sequence concave admissable function contracting linear operator convergence results convex admissable function convex hull convexity in measure Convexity Properties Definition ergodic theory extremal relative f and g f in measure family of functions family Q(T Fatou's lemma finite measure fjlx fnllx following property function f function X g e Q implies f integrable K X(x Let f Let Q llf^lL llfllx martingale theory measurable set measure space minimum X-norm neighborhood system norm Professor Errett Bishop proofs are omitted Q.E.D. Theorem real numbers Rn(f Rn(f)3 converge rotund satisfies the following satisfies the supremum sequence extracted sequence f space with X-norm subadditive submartingale subset sup f Suppose supremum condition theorem 13 Theorem 21 topology triangle inequality uniform uniformly convex uniformly X-integrable vector space