## Convergence Results as a Consequence of Convexity Properties of Certain Spaces |

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absolutely continuous additional condition admissable concave function admissable convex function Apply lemma bounded function bounded variation C-algebra Cauchy Cauchy sequence concave admissable function contracting linear operator convergence results convex admissable function convex hull convexity in measure Convexity Properties Definition denote derivative ergodic theory extremal relative f and g f in measure fall family G(T family of functions Fatou's lemma finite measure fnllx following property function and let function f Furthermore g e Q implies f Let f Let Q martingale theory measurable set measure space minimum M-norm neighborhood system non-negative norm Oftel positive functions Professor Errett Bishop proofs are omitted real numbers Rn(f rotund San Diego satisfies the following satisfies the supremum sequence extracted sequence f space Q subadditive submartingale sup f supermartingale supremum condition theorem 13 theorem 30 topology triangle inequality uniform convexity uniformly convex uniformly X-integrable