 Solution

ID: past-exam-of-the-probability-measure-and-martingales-course-of-the-university-of-oxford/2021/2/b/i/solution

For each

N \geq 1

, put

C_{N} = ⋃_{n \geq N} A_{n}

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}) .

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