Continuity of Probability

Definition

Continuity of Probability

Let be a probability space and let be a monotone sequence of events with limit . Then

Continuity from below

For a monotone increasing event sequence,

Continuity from above

For a monotone decreasing event sequence,

Proof

Proof

Increasing events: count each outcome when it first enters

Suppose . Set and define the newly added portions

If , then , whereas excludes . Thus .

Each outcome in enters at some first index , so

By the probability of a disjoint union and countable sigma-additivity,

The infinite sum is the limit of its partial sums. Hence

Decreasing events: pass to complements

Suppose . Their complements increase, and De Morgan’s law gives

The increasing case therefore gives . Using the probability of a complementary event,