In this paper, we study gauge norms on probability spaces and their associated Lebesgue spaces, both in the scalar and vector-valued cases. We consider norms that are symmetric with respect to groups of measure-preserving transformations, and we show that such a group is ergodic if and only if every symmetric gauge norm dominates parallel to parallel to(1). When the probability space is a compact group G with Haar measure mu, we study convolution of Banach algebra-valued functions in Lebesgue spaces. When G is Abelian and its dual group is linearly ordered, we study the associated Hardy spaces. When G = T, we characterize the closed densely defined operators on H-alpha (T) affiliated with H-infinity (T).
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