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Continuing my work through Folland, trying to prove the following (Chapter 7 #22):

Added: *Let $X$ be a locally compact Hausdorff space.*

Let $\{f_\alpha\}_{\alpha\in A}$ be a subset of $C_0(X)$ and $\{c_\alpha\}_{\alpha\in A}$ be a family of complex numbers. If for each finite set $B\subset A$ there exists $\mu_B\in M(X)$ such that $\|\mu_B\|\le 1$ and $\int f_\alpha\ d\mu_B=c_\alpha$ for $\alpha\in B$, then there exists $\mu\in M(X)$ such that $\|\mu\|\le 1$ and $\int f_\alpha\ d\mu=c_\alpha$ for all $\alpha\in A$.

Work so far: I think the idea is to use the result of Proposition 7.19, which says that if we have a sequence $\mu,\mu_1,\mu_2,\dots \in M(\mathbb{R})$, such that $\sup_n \|\mu_n\|<\infty$ with $F_n(x):=\mu_n((-\infty,x])\to F(x):=\mu((-\infty,x])$ at all $x$ where $F$ is continuously, then $\mu_n\to \mu$ vaguely.

My rough idea is that it suffices to assume that $A$ is countable, since we can use density arguments, and try to construct such a sequence satisfying 7.19, but I don't know where to go from there, let alone how to get there.

Any help would be greatly appreciated.

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What do we know about $X$? And it should be "$\int f_{\alpha}d\mu_B$ for $\alpha \in B$" instead of "$\int f_{\alpha}d\mu$ for $\alpha \in B$". –  Davide Giraudo Mar 25 '12 at 20:41
Thank you, Davide. I made the required edits. We can assume that $X$ is LCH. –  Eric Gregor Mar 25 '12 at 21:20
I've note thought a lot, so it may be wrong, but some arguments of weak-$*$ compactness of the unit ball of $M(X)$. –  Davide Giraudo Mar 25 '12 at 21:31

1 Answer 1

up vote 2 down vote accepted

Davide's suggestion is good.

Let $I$ be the collection of all finite subsets of $A$, ordered by inclusion. Verify that $I$ is a directed set. Then $\{\mu_B\}_{B \in I}$ is a net which is contained in the closed unit ball of $M(X)$. By Alaoglu's theorem it has a subnet, call it $\{\mu_{B_j}\}_{j \in J}$, where $J$ is some other directed set, which converges vaguely to some $\mu$. Now show that $\mu$ has the desired properties. (This will be good practice in working with nets. In particular, be very careful that you use the right definition of "subnet".)

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Thank you, I think I was lacking comfort with the notion of nets and completely skipped over Alaoglu's theorem. –  Eric Gregor Mar 26 '12 at 2:45
@EricGregor: If you really want to avoid nets, you could probably also use an equivalent notion of compactness: that the set $\{\mu_B\}$, being infinite (except in trivial cases), must have a vague limit point. –  Nate Eldredge Mar 26 '12 at 3:12

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