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In a general measure space $(X,\mathcal L,\lambda)$, a characterization ofweakly null sequences in $L_\infty (X,\mathcal L,\lambda)$ ($u_k\rightharpoonup 0$) in terms of their pointwise behaviour almost everywhere isderived from the Yosida-Hewitt identification of $L_\infty (X,\mathcalL,\lambda)^*$ with finitely additive measures, and extreme points of the unitball in $L_\infty (X,\mathcal L,\lambda)^*$ with $\pm \mathfrak G$, where$\mathfrak G$ denotes the set of finitely additive measures that take onlyvalues 0 or $ 1$. When $(X,\tau)$ is a locally compact Hausdorff space withBorel $\sigma$-algebra $\mathcal B$, the well-known identification of$\mathfrak G$ with ultrafilters means that this criterion for nullity isequivalent to localized behaviour on open neighbourhoods of points $x_0$ in theone-point compactification of $X$. Notions of weak convergence at $x_0$ and theessential range of $u$ at $x_0$ are natural consequences.When a finitelyadditive 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 andthose $\nu$ for which $\hat \nu$ is singular with respect to $\lambda$ arecharacterized.

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