# Problem to understand Borel-Cantelli Lemma

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!

• 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. – Dzoooks Mar 24 at 2:52
• It's not clear why the second assertion is true, if $A_n = A$. – Arctic Char Mar 24 at 2:59
• in second equation , is need event be independent? – masoud Mar 24 at 3:13
• I don't know if this answer might be helpful. – robjohn Mar 24 at 3:17
• 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 – 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
• Waht is $Th$ in your first paragraph? – M.Gonzalez Mar 24 at 19:51