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

Abstract

In a general measure space $(X,\mathcal L,λ)$, a characterization of weakly null sequences in $L_\infty (X,\mathcal L,λ)$ ($u_k \rightharpoonup 0$) in terms of their pointwise behaviour almost everywhere is derived from the Yosida-Hewitt identification of $L_\infty (X,\mathcal L,λ)^*$ with finitely additive measures, and extreme points of the unit ball in $L_\infty (X,\mathcal L,λ)^*$ with $\pm \mathfrak G$, where $\mathfrak G$ denotes the set of finitely additive measures that take only values 0 or $ 1$. When $(X,τ)$ is a locally compact Hausdorff space with Borel $σ$-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 $ν$ represents $f \in L_\infty(X, \mathcal B, λ)^*$ and $\hat ν$ is the Borel measure representing $f$ restricted to $C_0(X,τ)$, a minimax formula for $\hat ν$ in terms $ν$ is derived and those $ν$ for which $\hat ν$ is singular with respect to $λ$ are characterized.