Localizing Weak Convergence in ${\mathbf{L}}_{\infty }$

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 $L_{\infty }(X,\mathcal{L},\lambda {)}^{\ast }$ with finitely additive measures, and extreme points of the unit ball in $L_{\infty }(X,\mathcal{L},\lambda {)}^{\ast }$ 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 $f\in L_{\infty }(X,\mathcal{B},\lambda {)}^{\ast }$ 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.

Theorem 3.6, Corollary 3.7 and Section 3.1 are significant developments

Cited by in corpus (1)

• On singular extensions of continuous functionals from C([0,1]) to variable Lebesgue spaces