p-uniform measures on linear spaces \((0\leq p<\infty)\) (Q1262403)

Let E be a locally convex space and \(E'\) be the dual space. Let \(\mu\) be a probability measure on E of weakly p-th order, \(\tau^ p_{\mu}\) be the \(L^ p(\mu)\)-metric considered on \(E'\) and \(\Lambda_ p(\mu)\) be the closure of \(E'\) in \(L^ p(\mu)\). For A with \(\mu (A)>0\), \(\mu_ A\) be \(\mu_ A(B)=\mu (A\cap B)/\mu (A)\). Then \(\mu\) is called p-uniform if \(\tau^ p_{\mu_ A}\) and \(\tau^ p_{\mu}\) are equivalent for each A with \(\mu (A)>0\). The p-uniformness was first introduced by \textit{R. M. Dudley} [Z. Wahrscheinlichkeitstheor. verw. Geb. 6, 129-132 (1966; Zbl 0303.28012)]. In this paper, several characterizations of p-uniformness are given as follows. \(\mu\) is p-uniform if and only if for each sequence \(\{x_ n'\}\) in \(E'\), \(\mu\) (x;\(\sum_{n}| x_ n'(x)|^ p<\infty)=1\) or 0 according as the series \(\sum_{n}\| x_ n'\|^ p_{L^ p}\) converges or not. \(\mu\) is p-uniform if and only if \(\mu\) is 0- uniform and \(\Lambda_ 0(\mu)\subset L^ p(\mu)\). Suppose that \(\mu\) is 0-uniform, then \(\mu\) is p-uniform \((1\leq p<2)\) if and only if \(\Lambda_ p(\mu)\) is of type p-stable, and for \(p>2\) if and only if \(\Lambda_ p(\mu)\) is isomorphic to a Hilbert space. The Hilbertian support problem of a p-uniform measure is also considered.