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

J F Toland

Published 2018-02-06Version 1

For a general measure space $(X, \sL, \l)$ the pointwise nature of weak convergence in $\Li$ is investigated using singular functionals analogous to $\d$-functions in the theory of continuous functions on topological spaces. The implications for pointwise behaviour in $X$ of weakly convergent sequences in $\Li$ are inferred and the composition mapping $u \mapsto F(u)$ is shown to be sequentially weakly continuous on $\Li$ when $F:\RR \to \RR$ is continuous. When $\sB$ is the Borel $\sigma$-algebra of a locally compact Hausdorff topological space $(X,\varrho)$ and $f \in L_\infty(X, \sB, \l)^*$ is arbitrary, let $\nu$ be the finitely additive measure in the integral representation of $f$ on $L_\infty(X, \sB, \l)$, and let $\hat \nu$ be the Borel measure in the integral representation of $f$ restricted to $C_0(X,\varrho)$. From a minimax formula for $\hat \nu$ in terms $\nu$ it emerges that when $(X,\varrho)$ is not compact, $\hat\nu$ may be zero when $\nu$ is not, and the set of $\nu$ for which $\hat \nu$ has a singularity with respect to $\l$ can be characterised. Throughout, the relation between $\d$-functions and the analogous singular functionals on $\Li$ is explored and weak convergence in $L_\infty(X,\sB,\l)$ is localized about points of $(X_\infty, \varrho_\infty)$, the one-point compactification of $(X,\varrho)$.