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I am trying to construct a family of probability measures satisfying the following properties:

  • $A$ is an uncountable index set;
  • for each $\alpha\in A$, $\mathbb P_{\alpha}$ is a probability measure on the unit interval (endowed with the Borel $\sigma$-algebra);
  • if $C$ is an countable subset of this interval, then $\mathbb P_{\alpha}(C)=0$ for each $\alpha\in A$;
  • there exist pairwise disjoint (!) Borel sets $(S_{\alpha})_{\alpha\in A}$ such that $P_{\alpha}(S_{\alpha})=1$ for each $\alpha\in A$.

Can such a construction be explicitly given? Or at least can its existence be proven non-constructively (via Zorn’s lemma, for example)?

Any hints would be greatly appreciated.

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    $\begingroup$ For every $\alpha$ in $(0,1)$, consider $P_\alpha$ the image on $[0,1]$ of the product Bernoulli measure with parameter $\alpha$ by the function $f$ defined on $\{0,1\}^\mathbb N$ by $f((x_n)_{n\geqslant1})=\sum\limits_{n=1}^\infty x_n2^{-n}$. Then, by the strong law of large numbers, $P_\alpha(S_\alpha)=1$ where $S_\alpha$ denotes the set of real numbers whose binary expansion contains an asymptotic proportion $\alpha$ of ones. The sets $S_\alpha$ are disjoint hence you are done. $\endgroup$ – Did Oct 15 '16 at 9:42
  • $\begingroup$ @Did This example was actually on the back of my mind. Thank you very much for pointing out it actually works. If you put it as answer, I will accept it. $\endgroup$ – triple_sec Oct 15 '16 at 17:58

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