Localizing Weak Convergence in $\boldsymbol{ L_\infty}$

Abstract

In a general measure space $(X,\mathcal L,\lambda)$, a characterization of weakly null sequences in $L_\infty (X,\mathcal L,\lambda)$ ($u_k \rightharpoonup 0$) in terms of their pointwise behaviour almost everywhere is derived from the Yosida-Hewitt identification of $L_\infty (X,\mathcal L,\lambda)^*$ with finitely additive measures, and extreme points of the unit ball in $L_\infty (X,\mathcal L,\lambda)^*$ with $\pm \mathfrak G$, where $\mathfrak G$ denotes the set of finitely additive measures that take only values 0 or $ 1$. When $(X,\tau)$ is a locally compact Hausdorff space with Borel $\sigma$-algebra $\mathcal B$, the well-known identification of $\mathfrak G$ with ultrafilters means that this criterion for nullity is equivalent to localized behaviour on open neighbourhoods of points $x_0$ in the one-point compactification of $X$. Notions of weak convergence at $x_0$ and the essential range of $u$ at $x_0$ are natural consequences.When a finitely additive measure $\nu$ represents $f \in L_\infty(X, \mathcal B, \lambda)^*$ and $\hat \nu$ is the Borel measure representing $f$ restricted to $C_0(X,\tau)$, a minimax formula for $\hat \nu$ in terms $\nu$ is derived and those $\nu$ for which $\hat \nu$ is singular with respect to $\lambda$ are characterized.