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I need to solve the following problem.

Let $X_{1},X_{2},\dots$ be random variables on $(\Omega, \mathcal{F}, P)$. Show that the set $A = \{\omega \in \Omega :X_n(\omega) \hspace{1mm}converges\}$ is in F and that there exists a F-measurable random variable X such that $X_{n}(\omega) \to X(\omega)$ for $\omega \in A$.

Thanks in advance.

teo theo
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    There is a nice solution here: http://math.stackexchange.com/questions/1515873/measurable-functions-and-convergence –  Nov 08 '15 at 14:17
  • Let $X(\omega)$ be the limit if $\omega\in A$ and e.g. let it take value $0$ otherwise. – drhab Nov 08 '15 at 14:26
  • @drhab Under this setting we cover the part of the exercise which says that there exists an $F$-measurable random variable such that $X_n(\omega) \to X(\omega)$? – teo theo Nov 12 '15 at 18:50
  • Yes. The main part consists on proving that $A$ is measurable. After that $X$ can be defined like that as a random variable. – drhab Nov 12 '15 at 19:38

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