TITLE: Is the power set of $\mathbb{R}$ measurable by Dirac point measure?
QUESTION [0 upvotes]: By Dirac measure I mean if $p$ is a point in $\Omega$ then $\delta_p(X)$ = 1 if $p \in X$, $0$ otherwise, $\forall X \in \Omega$.
My instinct tells me this measure will work for the power set of $\mathbb{R}$---I can't come up with a scenario where there's ambiguity about whether $p$ is in or out---but I know that weirdness happens when uncountable sets are involved and so I'm not sure I can trust my instinct.
If I'm correct then a simple ye will do. If not, a counter example would be great.

REPLY [2 votes]: If $S$ is any non-empty set and $p \in S$ define $\delta_p(X)=1$ if $p \in X$ and $0$ otherwise. This is defined for all subsets $X$ of $S$ and it is a measure on the power set of $S$.  In particular, we can take $S=\mathbb R$.
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