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Localizing Weak Convergence in $\boldsymbol{ L_\infty}$. (arXiv:1802.01878v1 [math.FA])
来源于:arXiv
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 $\nu$ f 查看全文>>