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?
Probability measure on $\mathcal{P}(\mathbb{R})$
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measure-theory
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0@Shahab: You mean perhaps the strongest form of choice for which the Lebesgue measure is a measure on all subsets? We know that this implies that this is consistent with the principle of Dependent Choice, and it immediately negates the following: The weak ultrafilter lemma for $\mathbb N$ (i.e. every ultrafilter on $\mathbb N$ is principal), therefore ultrafilter lemma/Boolean Prime Ideal theorem; Axiom of choice for families of size $\aleph_1$, and therefore $\mathrm{DC}_{\aleph_1}$. – 2012-09-22
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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 $x\in I$.