Limit relation for events

Krizalid

MHF Hall of Honor
Mar 2007
3,656
1,699
Santiago, Chile
Let \(\displaystyle (\Omega,\mathcal F,P)\) be a probability space and \(\displaystyle (A_n)_{n\ge1}\) be a sequence of events in \(\displaystyle \mathcal F.\) Prove that \(\displaystyle {{I}_{\underset{n\to \infty }{\mathop{\underline{\lim }}}\,{{A}_{n}}}}+\underset{n\to \infty }{\mathop{\overline{\lim }}}\,{{I}_{A_{n}^{c}}}=1.\)

(Rock)
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey Krizalid.

What is I exactly? Is it some sort of indicator function?
 

Krizalid

MHF Hall of Honor
Mar 2007
3,656
1,699
Santiago, Chile
Yes.
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
What event are you setting the indicator functions for? Also what does the line about the second limit mean?

Intuitively if you have some event E1 with an indicator that returns 1 if you have the event and 0 if not then I(E1) + I(not E1) should equal 1 by all intuitive given that one event is entirely complementary to the other.

I'm guessing you are going to have use the property of subsets in a filtration. Correct me if I am wrong but the limit of the sequence in a filtration should get smaller and smaller since the later events in the sequence should be a subset of the earlier ones.

If the above is the case then the limit should go to an event space with a simple event.

Do you have any definitions of what sets the indicator on or off?