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Given a pobability space $(\Omega,\mathcal{A},P)$, a filtration $(\mathcal{F_n})_n$ and a sequence of events $A_n \in \mathcal{F}_n$

I have the martingale $X_n = \sum_{1 \leq j \leq n}(\mathbb{1}_{A_j} - P[A_j|\mathcal{F}_n])$ and I know that $P[B] = 1$, with $B = \{\limsup_nX_n$ exists and is finite$\}$ $\cup$ $\{\limsup_nX_n = \infty, \liminf_nX_n = -\infty\}$

I want to show for $\omega \in B$, that $\sum_{n \geq 1}\mathbb{1}_{A_n}(\omega) = \infty$ iff $\sum_{n \geq 1} \mathbb{E}[\mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \infty$

So, I want to exchange a limit and the integral, to achieve:

$$\sum_{n \geq 1} \mathbb{E}[\mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \lim_N \sum_{1 \leq n \leq N} \mathbb{E}[\mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \lim_N\mathbb{E}[\sum_{1 \leq n \leq N} \mathbb{1}_{A_n}|\mathcal{F}_{n-1}](\omega) = \mathbb{E}[\sum_{n\geq 1}\mathbb{1}_{A_n}|\mathcal{F}_{n-1}]$$

Now my question is, how can I argue, that I can pull the limit into the expectation, if the limit is infinite? Because if the limit were finite, then I could use monotone convergence... but I don't know what to do it it is infinte...

Any help is appreciated :)

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  • $\begingroup$ Duplicate of this question $\endgroup$
    – saz
    Nov 29, 2019 at 17:57
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    $\begingroup$ The Monotone Convergence Theorem doesn't require the limit to be finite. It perfectly works with infinite limit. Note that this limit will always exist in the extended reals because the sequence is monotonically increasing. $\endgroup$ Nov 30, 2019 at 13:13
  • $\begingroup$ Does this answer your question? Generalized Second Borel-Cantelli lemma $\endgroup$
    – Math1000
    Jan 19, 2020 at 21:51

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