If $X, X_1, X_2, \ldots$ are real random variables defined on a probability space $(\Omega, \mathcal{A}, \mathbf{P})$, we say $X_n$ converges almost surely to $X$, if $$\mathbf{P}\Bigl[\{\omega\in\Omega : \lim_{n\rightarrow\infty} X_n(\omega) = X(\omega) \}\Bigr] = 1\, .$$

Now assume the random variables $X_1, X_2, \ldots$ converge pointwise to some random variable $Z$. Does the following hold $$\mathbf{P}\bigl[\{\omega \in \Omega : Z(\omega) = X(\omega)\}\bigr] = \mathbf{P}\Bigl[\{\omega\in\Omega : \lim_{n\rightarrow\infty} X_n(\omega) = X(\omega) \}\Bigr] \, ?$$

Thanks for any help!

  • 3
    $\begingroup$ Yes. Except from a null set, $Z=\lim X_n =X$. $\endgroup$ – Brian Ding Feb 18 '15 at 2:26

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