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Let $(X,\mathfrak B)$ be a measurable space such that $\{x\}\in \mathfrak B$ for all $x\in X$, and let $\mu$ be a positive measure on this space such that $$ \mu(B) \in\{0,1\} \quad\text{for all }B\in \mathfrak B. $$ What are the mildest conditions on $(X,\mathfrak B)$ that imply that $\mu =\delta_x$ for some $x\in X$?

It is known to hold for $\Bbb R$ with a Borel $\sigma$-algebra, and I believe it fairly easy extends to $\Bbb R^n$. I wonder, though, whether it holds at least for locally compact Polish spaces, or perhaps for more general case. I am also interested in examples of spaces where such statement does not hold.

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I guess you mean $\{x\}\in \mathfrak B$. But it is not necessary that singletons are measurable to define Dirac measures. –  Byron Schmuland Jan 31 '13 at 18:10
    
@ByronSchmuland: do you mean that such Dirac measures are defined over general atoms? –  Ilya Jan 31 '13 at 18:31
    
For $B\in\mathfrak B$, let $\delta_x(B)=1_{[x\in B]}$. The definition does not need singletons to be measurable. –  Byron Schmuland Jan 31 '13 at 18:34
    
@ByronSchmuland: Indeed, thanks for this point. –  Ilya Jan 31 '13 at 18:56
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2 Answers 2

up vote 3 down vote accepted

An easy example where it does not hold is an uncountable set endowed with the countable-cocountable $\sigma$-algebra. If one let's $\mu(B)=1$ if $B$ has countable complement and $\mu(B)=0$ otherwise, one gets an example where the measure is not a Dirac measure.

On a countably generated space, every $0-1$-valued measure is on an atom of the $\sigma$-algebra and hence a Dirac-measure. This is shown in Borel Spaces by Rao and Rao, on page 14. The basic idea is to take a countable sequence $C_1,C_2,\ldots$ of generators and define $D_n=C_n$ if $\mu(C_n)=1$ and $D_n=C_n^c$ otherwise. Then $D=\bigcap_n D_n$ is an atom of the $\sigma$-algebra such that $\mu(D)=1$. If $x\in D$, we have $\delta_x=\mu$.

In particular, the result holds for all Polish spaces.

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Nice! Let me check Rao and Rao –  Ilya Jan 31 '13 at 18:07
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A nice case to consider:

Let $X$ be a completely regular topological space. The zero sets in $X$ are sets of the form $\{x \in X : f(x)=0\}$ where $f : X \to \mathbb R$ is continuous. The sigma-algebra $\mathcal B$ of Baire sets in $X$ is the sigma-algebra generated by the zero sets. Alternatively, $\mathcal B$ is the smallest sigma-algebra on $X$ such that every continuous real-valued function is $\mathcal B$-measurable.

The condition:

every (countably-additive) $\{0,1\}$-valued measure on $\mathcal B$ is a Dirac measure

is equivalent to the condition that $X$ is real-compact.

References:
Gillman & Jerison, Rings of Continuous Funtions.
W. Moran, "Measures and mappings on topological spaces." Proc. London Math. Soc. 19 (1969) 493-508.

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The condition is theorem 16 in Hewitt, Linear functionals on spaces of continuous functions, Fund. Math. 37 (1950), 161-189. matwbn.icm.edu.pl/ksiazki/fm/fm37/fm37113.pdf –  Martin Jan 31 '13 at 19:27
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