Supports of measures with weak moments (Q1945205)

Let \( \mu \) be a probability measure on a separable Frechet space \(X\) with dual \(X^{*} \subset L^{p}(\mu) \; (p \geq 1)\) (weak moment of order \(p\)). A reflexive, separable Banach \(E\) is said to be compactly embedded in \(X\) if \(E\) is a linear subspace of \(X\) with norm \(\| . \|_{E} \) with respect to which it is reflexive and separable and its unit-ball is precompact in \(X\). \(E\) is said to be of full \(\mu\)-measure if \(\mu(E)=1\). For \(g \in L^{q}(\mu)\) (\(q\) being the conjugate of \(p\)), \(F_{g}\) is the functional on \(X^{*}\) defined by \( F_{g}(f)= \int gf d \mu\) (this functional is denoted by \(F_{A}\) if \(g = \chi_{A}\) for some measurable set \(A)\); this gives an operator \(h: L^{q}(\mu) \to X\), \(\langle h(g), f\rangle = F_{g}(f) \; \forall f \in X^{*} \). If there is a point \(m \in X\) such that \( \int f d \mu = f(m) \; \forall f \in X^{*}\), then \(m\) is called the mean of the measure \(\mu\). When \(p=2\), then a covariant operator \(K: X^{*} \to X\) is defined by \(\langle f, K(g)\rangle = F_{g}(f)\). For these measures, the author proves some results about the existence of separable and reflexive Banach spaces \(E\) with \(E^{*} \subset L^{p}(\mu)\) and, in some cases, with full \(\mu\)-measure. Some of the main results are the following. I. Suppose \(p=1\). Then there is a compactly embedded, reflexive and separable Banach space \(E\) with a full \(\mu\)-measure and \(E^{*} \subset L^{1}(\mu)\) if and only if, for each \(\mu\)-measurable set \(A\), there exists \(h_{A} \in X\) such that \( F_{A}(f)= f(h_{A}) \; \forall f \in X^{*}\). II. Assume \(p > 1\). Then there is a compactly embedded, reflexive and separable Banach space \(E\) with \(E^{*} \subset L^{p}(\mu)\) if and only if the set \( H= \{ h(g): g \in L^{q}(\mu), \| g \|_{L^{q}(\mu)} \leq 1 \} \) is precompact in \(X\). III. Let \(p = 2\). Then there is a compactly embedded, reflexive and separable Banach space \(E\) with a full \(\mu\)-measure and \(E^{*} \subset L^{2}(\mu)\) if and only if the set \( K(S)\) is precompact in \(X\) for every absolutely convex \(\sigma(X^{*}, X)\)-compact set \(S \subset X^{*}\).