Continuity Theorem of Probability

Asked Apr 24 '20 at 17:26

Active Apr 26 '20 at 12:27

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I came across this Theorem in Introduction to Mathematical Statistics and I don't understand what it is stating:

Let ${C_n}$ be a nondecreasing sequence of events. Then $$\lim_{n\to\infty} P(C_n) = P(\lim_{n\to\infty} C_n) = P(\bigcup\limits_{n=1}^\infty C_n).$$

P is a probability function here. I understand the statement that $\lim_{n\to\infty} C_n = \bigcup\limits_{n=1}^\infty C_n$ for a nondecreasing sequence of events, but that is a result of set theory. What is the significance of interchanging P and the limit in the theorem above?

edited Apr 24 '20 at 20:34

William Elliot

17,598

asked Apr 24 '20 at 17:26

J. Grohmann

1

Its a property of measures known as continuity from below. – Muselive Apr 24 '20 at 18:14
https://math.stackexchange.com/questions/234292/continuity-from-below-and-above – Muselive Apr 24 '20 at 18:14
You want a proof of it or you want why interchanging limit and P is important? – Masoud Apr 26 '20 at 12:31

Continuity Theorem of Probability

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