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I consider a standard normal random variable $X$ and a Vitali set $V$. $P(X\in V)$ can not be computed as $V$ is not measurable.

Now I consider the outcome of the following experiment $E_S$ : $N_S$ is the number of experiments $X_i$ (all independent and equals to $X$, and $i\in\mathbb N$) such that $X_i\in S$.

  1. If $P(X\in S)=0$ then $P(N_S=0)=1$
  2. If $P(X\in S)>0$ then $P(N_S=\infty)=1$
  3. What happens for $S=V$ ? I think that $P(N_V=\infty)=0$ and $P(N_V=0)<1$. Am I right and can we obtain some more precise results ?

Thank you for your answers !

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    thanks for the english lesson :)2012-08-01

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It is a bit surprising to see that this post is still alive... but here we go. The question is:

What happens for $S=V$?

In a nutshell, and as was already explained in the comments, what happens is that nothing guarantees that the sets $[N_V=\infty]$ or $[N_V=0]$ are measurable, hence neither $P(N_V=\infty)$ nor $P(N_V=0)$ is defined. Thus, asking if these probabilities are $0$ or $\lt1$ or whatever has no sense.

Let us recall why (the function) $N_S$ is measurable when (the subset) $S$ is measurable. One writes $ N_S=\sum_{i=1}^{+\infty}\mathbf 1_{A^S_i},\qquad A^S_i=[X_i\in S], $ and, by the definition of the measurability of $X_i$ and $S$, each set $A^S_i$ is measurable hence each function $\mathbf 1_{A^S_i}$ is measurable and, by measurability of pointwise limits, the function $N_S$ is measurable.

When $V$ is not measurable, the reasoning above breaks down at the moment when one needs each $A^V_i$ to be measurable. For example, $ [N_V=0]=\bigcap_{i=0}^{+\infty}(\Omega\setminus A_i^V)=\bigcap_{i=0}^{+\infty}[X_i\notin V], $ and none of the subsets $[X_i\notin V]$ is measurable, a priori. If you find a way to prove that these subsets are in fact measurable, or only that their whole intersection is measurable (something which could happen without every $[X_i\notin V]$ being measurable), please go ahead. Otherwise...

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    Ok, I got your point, but could you please give me some details about "the unmentioned aspects of the setting" ? What do I need to make it possible ? Perhaps just a link to some explanations ?2012-08-01