Past exam of the probability, measure and martingales course of the University of Oxford / 2021 / 2 / b / i

Table of contents
- Solution i

Solution

For each

N \geq 1

, put

C_{N} = ⋃_{n \geq N} A_{n}

. Closure of a sigma-algebra under countable unions and countable intersections gives

A = lim sup_{n \to \infty} A_{n} = ⋂_{N \geq 1} C_{N} \in F .

C_{N}

decrease to

A

. Continuity of probability from above applies because

P (C_{1}) \leq 1 < \infty

. Also,

A_{n} \subseteq C_{N}

whenever

n \geq N

, so

P (C_{N}) \geq sup_{n \geq N} P (A_{n}) .

Letting

N \to \infty

proves

P (A) = N \to \infty lim P (C_{N}) \geq N \to \infty lim n \geq N sup P (A_{n}) = n \to \infty lim sup P (A_{n}) .

This reverse Fatou inequality for events requires no independence.

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version