Continuity of Probability Measures

Ponder...

For an increasing sequence in with , the numbers increase to ; for a decreasing sequence with , they decrease to .

Proposition 2.2 (Continuity of Probability Measures 1)

Let where and for all . We call an increasing sequence in .

Then is an increasing sequence in and converges to .

Decreasing sequences

Proposition 2.3 (Continuity of Probability Measures 2)

Let where and for all . We call a decreasing sequence in .

Then is a decreasing sequence in and converges to .

For an increasing sequence, define and for . Then is a disjoint collection with , so countable additivity gives . This converges to .

For a decreasing sequence, the complements form an increasing sequence in with union . Applying the increasing case and using ,