I needed to prove the Borel-Cantelli lemma, which states that

If $$ \sum_{n=1}^{\infty} P(A_n) < \infty \implies P(\limsup A_n) = > 0,$$

$$\sum_{n=1}^{\infty} P(A_n) = \infty \implies P(\limsup A_n) = 1 $$

I could understannd the proof, but I am struggling with a detail:

If this probability is defined in a probability space, say $(\Omega, \mathbb{F}, P) $, then $ \bigcup_{n=1}^{\infty} A_n \in \mathbb{F} $. Then how is the following equality possible? $$ \sum_{n=1}^{\infty} P(A_n) = \infty $$

I believe it is due to the fact that $$ P\left(\bigcup_{n=1}^{\infty}A_n\right) \leq \sum_{n=1}^{\infty} P(A_n) $$

which makes possible for $ \bigcup_{n=1}^{\infty} A_n \in \mathbb{F} $ and still the sum of the probabilities not to be convergent. But it's still quite misty to me, I can't imagine an example where the sum of the probabilities would not be convergent. Could you give me one example?

Thank you!

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    $\begingroup$ Just set $A_n = A$ for all $n$ and take $A$ such that $P(A) > 0$. For example, let $A$ be the event of rolling a 1 on a 6-sided die. $\endgroup$ – Dzoooks Mar 24 at 2:52
  • $\begingroup$ It's not clear why the second assertion is true, if $A_n = A$. $\endgroup$ – Arctic Char Mar 24 at 2:59
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    $\begingroup$ in second equation , is need event be independent? $\endgroup$ – masoud Mar 24 at 3:13
  • $\begingroup$ I don't know if this answer might be helpful. $\endgroup$ – robjohn Mar 24 at 3:17
  • $\begingroup$ let toss a coin, $\{A_n\}_{n\geq 1}$ $A_n$ is the outcome of coin that is $H$ or $T$. $A_n$ are independent $\endgroup$ – masoud Mar 24 at 3:19

let $\{A_n,n\geq 1\}$ is a sequence of tossing a coin. define $A_n$ that we observe $\{H \}$ in n-Th .

$ lim sup A_n$ ,That means the event that observe infinitely many of $H$,. so and $P(lim sup A_n)=1$ .

according to lemma $\sum p(A_n)=\infty$ and the results are same.

let $B_n$ be that not any of $A_1,\cdots ,A_n$ be "H". $P(lim sup B_n)=0$

also $\sum (\frac{1}{2^n}) < \infty$ so the results are same

  • $\begingroup$ Waht is $Th$ in your first paragraph? $\endgroup$ – M.Gonzalez Mar 24 at 19:51
  • $\begingroup$ i am not good in english. i means, the resulte of the cion tossing in the experiment with number n. like 5th $\endgroup$ – masoud Mar 24 at 20:20

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