Probability Function

up:: 023 MOC Measure Theory

A probability function is a function defined from the Sigma-algebra of a Measurable Space to the real numbers, which satisfies

• ;
• For a countable set of mutually disjoint measurable sets , we have

Properties

• ¹
• ²
• ³
• Continuity Theorem of Probabilities over (non-)decreasing sets of events

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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. Since , and since . ↩

2. Note that of disjoint sets; thus, . ↩

3. Note that and is a union of disjoint parts, from which . ↩