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Let us consider the space of Borel, regular, complex measures on the real line, endowed with the total variation norm. Inside this space, I would like to characterize the space of all the finite complex linear combinations of Delta measures. In particular, could we approximate a measure with compact support with linear combinations of that kind? Maybe it is possible only in the weak star topology... Thank you

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2 Answers 2

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You can also "just" apply Hahn-Banach. The dual space of $(C_0(\mathbb R)^*,w^*)$ is $C_0(\mathbb R)$. So, to prove that the linear combinations of Delta measures are $w^*$-dense it is enough to check that if a function $f\in C_0(\mathbb R)$ is such that $\int f d\delta_a=0$ for every $a\in\mathbb R$ then $f=0$; but this is obvious since $\int f d\delta_a=f(a)$.

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  • $\begingroup$ this is even more nicier! $\endgroup$
    – Norbert
    Sep 10, 2013 at 22:46
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    $\begingroup$ You're welcome! $\endgroup$
    – Etienne
    Sep 11, 2013 at 16:37
  • $\begingroup$ This should word with $\mathbb{R}$ replaced by any locally-compact space no? $\endgroup$
    – ABIM
    Jul 31, 2019 at 11:40
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Dirac measres are extreme points of the convex set of subprobability measures $P\subset C_0(\mathbb{R})^*$. This set is compact in the weak-$^*$ topology, so by Krein-Milman theorem every probability measure is a weak-$^*$ limit of finite convex combination of Dirac delta measures. For details see example 8.16 here.

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  • $\begingroup$ The set of subprobability measures in $C_0(\mathbb{R})^*$ is weakly$^*$ compact; that of probability measures is not closed, $\delta_n \to 0$. $\endgroup$
    – yada
    Nov 14, 2019 at 20:00
  • $\begingroup$ @yadaddy, Fixed $\endgroup$
    – Norbert
    Nov 14, 2019 at 20:59

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