I've got almost not background in measure theory and just scratched the surface of distribution theory, but I've been manipulating quite a lot of stats and probabilities recently until this question popped in my mind: is there any real variable with zero variance that is not a dirac delta or which measure in not equivalent to $d\mu = \delta_{\langle X\rangle}dX$?
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Assuming $X$ does have moments up to order $2$, then if $\operatorname{Var}X=0$ it must be the case that $X=a$ a.s. (for some constant $a\in\mathbb{R}$). Indeed, $$\operatorname{Var}X= \mathbb{E}\left[(X-\mathbb{E} X)^2\right]$$ and as the r.v. $(X-\mathbb E X))^2$ is non-negative, this implies $(X-\mathbb{E} X)^2=0$ a.s. So $X-\mathbb E X=0$ a.s. Setting $a\stackrel{\rm{}def}{=}\mathbb E X$, you have $\mathbb{P}_X(\{a\})=1$.
Clement C.
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This answer is mostly correct, but please note that a.e. should be a.s.. a.e. is related, but incorrect here. – JPi Jan 28 '14 at 00:48
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Indeed — sorry. – Clement C. Jan 28 '14 at 00:52
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No need to apologize! – JPi Jan 28 '14 at 00:52