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On a metric space $X$, Did said:

the space $M_1$ of probability measures is included in the dual of the space $C_b$ of bounded continuous functions.

I was wondering what is the representation theorem for the dual of $C_b(X)$, which is supposed to have $M_1$ as its subset?

I saw in Wikipedia, only Riesz representation theorems for $C_0$ and $C_c$, not for $C_b$.

Thanks and regards!

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No representation theorem needed. Just integrate with respect to $\mu \in M_1$ to get a continuous linear functional on $C_b(X)$. –  Martin Feb 21 '13 at 3:16
    
@Martin: Thanks! (1) In that scenario, is $C_b(X)$ endowed with the the supremum norm? (2) Can the continuous dual of $C_b(X)$ be represented as a set of measures which contains $M_1$? –  Tim Feb 21 '13 at 3:22
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(1) Yes. (2) No. –  Martin Feb 21 '13 at 3:23
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You edited your question (2)... For the new (2): $C_b(X) = C(\beta X)$ and apply the Riesz representation theorem for $\beta X$, so it is a set of measures, but not measures on $X$. –  Martin Feb 21 '13 at 3:28
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I'm not a person to say what Did can and what not. Anyway, he already said that, and that's true. Even more, $\mathcal M_1$ is a linear subspace of a continuous dual of $\mathrm b\mathfrak B_X$ - the Banach space of bounded measurable function for any measurable space $(X,\mathfrak B_X)$ endowed with a $\sup$-norm. To state that $\mathcal A$ is a subset of a continuous dual of $\mathcal B$ you only have to check that any element of $\mathcal A$ acts continuously on $\mathcal B$, the actual dual $\mathcal B^*$ does not have to be computed. –  Ilya Feb 22 '13 at 17:13

1 Answer 1

up vote 2 down vote accepted

For the sake of having an answer to the question, Martin's comment contains the key point. We have an isomorphism $C_b(X) \cong C(\beta X)$ where $\beta X$ is the Stone-Čech compactification (defined for arbitrary topological spaces; some authors put extra conditions on $X$ but these are just the conditions required for the natural map $X \to \beta X$ to be an embedding), so by the ordinary Riesz representation theorem describes the dual of $C_b(X)$ in terms of measures on $\beta X$.

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Thanks! But still not the same as what Did say. He said the probability measures on $X$ instead of $\beta X$ is a subset of dual of $C_b(X)$. For that purpoe, shall the measures on $βX$ be restricted to $X$ (we can do this because X is metric space and thus Tychonoff, the mapping from X to its image in βX is injective and homeomorphism)? –  Tim Feb 22 '13 at 21:30
    
As Martin has already said, you don't need a representation theorem to see this. Just integrate measures on $X$ against functions in $C_b(X)$. –  Qiaochu Yuan Feb 22 '13 at 21:31

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