Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Earlier I had posted question about Borel cantelli Lemma. Thanks to Nate and Shai for your kind response. Please see my profile for the earlier questions. As you can see I do not understand the lemma well. Here are some confusions I have. I hope someone will be kind enough to help me understand the following:

Is it true that P ([An i.o.]∩ A) = P [( A ∩ An) i.o.]. If so, how do we prove this? or is it trivial? And assuming that we know that P[An i.o.] =1 , can we establish that the sum {P(A ∩ An)} diverges (meaning = infinity) ?


share|cite|improve this question

The events "$A_n$ for infinitely many $n$; and $A$" and "$A_n \land A$ for infinitely many $n$" are one and the same since $\land$ distributes over $\lor$: $$ A \land \bigwedge_{n \geq 0} \bigvee_{m \geq n} A_m \Leftrightarrow \bigwedge_{n \geq 0} A \land \bigvee_{m \geq n} A_m \Leftrightarrow \bigwedge_{n \geq 0} \bigvee_{m \geq n} A \land A_m.$$

As for your second question, if $A$ never happens then $P(A \land A_n) = 0$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.