Theorems of Limits in a Probability Measure

Topics: Probability Measure - Probability

---

(theorem)

Let $(A_n{)}_{n\ge 1}$ be a non-decreasing sequence of events. Then:

$$\mathbb{P}\left(\bigcup _{n=1}^{\infty }{A}_n\right)=\underset{n\to \infty }{\lim }\mathbb{P}(A_n)$$

…where ${\bigcup }_{n=1}^{\infty }{A}_n={\lim }_{n\to \infty }{A}_n$ (the limit of the sequence).

(theorem)

Let $(A_n{)}_{n\ge 1}$ be a non-increasing sequence of events. Then:

$$\mathbb{P}\left(\bigcap _{n=1}^{\infty }{A}_n\right)=\underset{n\to \infty }{\lim }\mathbb{P}(A_n)$$

…where ${\bigcap }_{n=1}^{\infty }{A}_n={\lim }_{n\to \infty }{A}_n$.

(theorem)

Let $(A_n{)}_{n\ge 1}$ be a sequence of events that converges to an event $A$. Then:

$$\mathbb{P}(A)=\underset{n\to \infty }{\lim }\mathbb{P}(A_n)$$