Continuity Theorem of Probabilities over (non-)decreasing sets of events

up:: Probability Function

Probabilities commute with limits taken with respect to non-decreasing and decreasing sets of events.

Proof for non-decreasing sets

Let be a non-decreasing sequence, i.e. , “converging” to . Then we seek to prove

For that, note that this sequence “grows radially” from . For that, define “rings” to be

1. With that, we see that ¹.
2. Note also that , and, thus, are all mutually exclusive.
3. Note that, since , we have that

Given all that, note that

Proof for decreasing sets

Let be a non-decreasing sequence, i.e. , “converging” to . Then we seek to prove

Note that , we have

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References

• Hogg, Robert V., Joseph W. McKean, and Allen T. Craig. Introduction to mathematical statistics. Pearson Education India, 2013.

Footnotes

1. Due to and incrementing with these “excesses” ↩