Generalized Lebesgue Spaces and Application to Statistics

Statistics requires consideration of the ``ideal estimates'' defined through the posterior mean of fractional powers of finite measures. In this paper we study , the linear space spanned by th power of finite measures, . It is shown that generalizes the Lebesgue function space , and shares most of its important properties: It is a uniformly convex (hence reflexive) Banach space with as its dual. These results are analogous to classical counterparts but do not require a dominating measure. They also guarantee the unique existence of the ideal estimate.