Prove $P\big(\lim (\inf A_n)\big) \le\big( \lim \inf P(A_n)\big)$

Question

Let we have $(A_n: n\ge 1)$ a succession of events in a probability space. I have to prove that:

1. $P\big(\lim (\inf A_n)\big) \le \lim \big(\inf P(A_n)\big)$
2. $\lim \big(\sup P(A_n)\big) \le P(\lim \big(\sup A_n\big)$
3. If there exists $\lim A_n$, then $P(\lim A_n)=\lim P(A_n)$

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• I don't know how to prove the first one.

• For the second one, I have done:

Using that $(\lim (\sup A_n)^c=\lim (\inf A_n^c)$ and 1.,

$P\big(\lim (\sup A_n)\big)=1-P\big(\lim (\inf A_n^c)\big) \ge 1-\lim \big(\inf P(A_n^c)\big)=1-\lim\big( \inf (1-P(A_n))\big)$

But I don't know why is that $\ge \lim \big(\sup P(A_n)\big)$.

• And for the third one I have used 1. and 2.:

If there exists $\lim A_n$, we know that $\lim inf A_n =\lim (A_n)=\lim (\sup A_n)$ and $\lim \big(\inf P(A_n)\big) \le \lim \big(\sup P(A_n)\big)$ is always true, so

$P(\lim A_n)=P\big(\lim (\inf A_n)\big) \le \lim \big(\inf P(A_n)\big) \le \lim P(A_n) \le \lim\big( \sup P(A_n)\big) \le P\big(\lim (\sup A_n)\big)=P(\lim A_n) \rightarrow P(\lim A_n)=\lim P(A_n)$

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So could anybody help me to prove the first one, please?

Answer 1

Beware that $$1_{\liminf A_n}=\liminf1_{A_n}\text{ and }1_{\limsup A_n}=\limsup1_{A_n}$$Then applying Fatou we find:$$P(\liminf A_n)=\int1_{\liminf A_n}=\int\liminf1_{A_n}\leq\liminf\int1_{A_n}=\liminf P(A_n)\tag1$$

Further we have:$$\limsup A_n=(\liminf A_n^{\complement})^{\complement}$$and applying $(1)$ we find:$$P(\limsup A_n)\geq1-\liminf (1-P(A_n))=-\liminf(-P(A_n))=\limsup P(A_n)$$

where the last equality is based on the general rule: $$\liminf (-c_n)=-\limsup c_n$$

Answer 2

Define $B_N:=\bigcup_{n=1}^N\bigcap_{k=n}^{+\infty}A_k$: we have $$P(\liminf A_n)=\lim_{N\to+\infty}P(B_N)=\lim_{N\to+\infty}P(\bigcap_{k=N}^{+\infty}A_k)\leq\liminf_{N\to+\infty}P(A_N).$$

In the first equality I used monotone convergence, in the second I used the fact that if $x\in\bigcap_{k=i}^{+\infty}A_k$ then $x\in\bigcap_{k=j}^{+\infty}A_k$ for every $j\geq i$ and in the last step I used $\bigcap_{k=N}^{+\infty}A_k\subseteq A_N$ and the monotonicity of the measure.