Probability of Limit of Sequence of Events/Decreasing

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\sequence {B_n}_{n \mathop \in \N}$ be a decreasing sequence of events.

Let $\ds B = \bigcap_{i \mathop \in \N} B_i$ be the limit of $\sequence {B_n}_{n \mathop \in \N}$.


Then:

$\ds \map \Pr B = \lim_{n \mathop \to \infty} \map \Pr {B_n}$


Proof


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Set $A_i = \Omega \setminus B_i$ and then apply De Morgan's laws and the result for an increasing sequence of events.

$\blacksquare$


Sources

  • 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 1.9$: Probability measures are continuous
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