DISCREPANCY NORMS ON THE SPACE M[0,1] OF RADON MEASURES

摘要

L~1 [0, 1] is the space of all Lebesgue integrable functions on [0, 1] and m denotes the Lebesgue measure on [0, 1]. We can see L~1 [0, 1] as a closed subspace of M[0, 1]. A measure μ in M[0, 1] is called diffuse if μ({x}) = 0 for each x in [0, 1]. Thispaper consists of two sections. In Section 1 we study the structure of the M[X] spaces. The papers [Wei] and [B] are cornerstones in our considerations. Theorem 4.2 in [Wei] about the M_0 space (related to Proposition 11 in [B]) can be extended to certain classes of M[X] spaces.This is the content of Theorem 1.1 and Theorem 1.5. More precisely in Theorem 1.1 we show that if X has a symmetric basis and contains no copy of l~1 then every diffuse measure μ in M[0, 1] is the limit in the M[X]-norm of a sequence (μ_n) of measures such that each μ_n is absolutely continuous with respect to the Lebesgue measure m on [0, 1]. Theorem 1.5 asserts the following.