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This question has been bugging me for a while? Does there exist a probability measure on the measurable space $\bigl(\mathbb{R},\mathcal{P}(\mathbb{R})\bigr)$. If so, what is it?

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Yes, $\delta_0$ for example. (Dirac at $0$). I guess you want additional conditions. – Davide Giraudo Sep 22 '12 at 11:43
As @DavideGiraudo said, you probably want more conditions. I am still looking for a natural measure (on $\mathcal{P}([0,1])$). Though I think I now know such an example. Let me know if you're interested in that. – Quinn Culver Sep 22 '12 at 12:42
To read about the space $\mathcal{P}(\mathcal{P}(\mathbb{R}))$, check out Billingsley's Convergence of Probability Measures and Parthasarathy's Probability Measures on Metric Spaces. – Quinn Culver Sep 22 '12 at 12:45
If you drop the axiom of choice then you can have the Lebesgue measure... – Asaf Karagila Sep 22 '12 at 13:19
@Quinn I think he means power set by ${\cal P}$, not the space of probability measures. – Byron Schmuland Sep 22 '12 at 14:49
up vote 6 down vote accepted

Reference: You may be interested in the "problem of measure". There is a short treatment of this topic in Appendix C of Real Analysis and Probability by R.M. Dudley.

He proves the following result due to Banach and Kuratowski: Assuming the continuum hypothesis, there is no measure $\mu$ defined on all subsets of $I:=[0,1]$ with $\mu(I)=1$ and $\mu(\{x\})=0$ for all $x\in I$.

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