Limit of an increasing sequence of probability measures is a probability

Question

Let {$P_{n}$} be a sequence of probabilities on a $\sigma$-field $F$ satisfying $P_{n}(A)$ $\le$ $P_{n+1}(A)$ for all $A$ and $n$. Define $P(A) = sup P_{n}(A)$ for each $A$ in $F$. Is $P$ a probability measure?

Answer 1

Recall that any monoton and bounded sequence is convergent. That is for any $A$ there exists a P(A) such that

$$\lim_{n\to \infty}P_n(A)=P(A)=\sup P_n(A).$$

We have to prove that $P$ is a probability measure on the measurable space $[\Omega,\mathcal A]$ on which the $P_n$s are probability measures. $0\le P\le1$ for any $A\in \mathcal A$ and $P(\emptyset)=0$. It remains to demonstrate that for any countable collection of pairwise disjoint sets $\{A_n\}$

$$P\left(\bigcup_{n=1}^{\infty}A_n\right)=\sum_{n=1}^{\infty}P(A_n).$$

Now

$$P\left(\bigcup_{n=1}^{\infty}A_n\right)=\lim_{m\to \infty}P_m\left(\bigcup_{n=1}^{\infty}A_n\right)=$$ $$=\lim_{m\to \infty}\sum_{n=1}^{\infty}P_m(A_n)=\sum_{n=1}^{\infty}\lim_{m\to \infty}P_m(A_n)=\sum_{n=1}^{\infty}P(A_n).$$