Sub-probability measure

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In the mathematical theory of probability and measure, a sub-probability measure is a measure that is closely related to probability measures. While probability measures always assign the value 1 to the underlying set, sub-probability measures assign a value lesser than or equal to 1 to the underlying set.

Let $\mu$ be a measure on the measurable space $(X,{\mathcal {A}})$ .

Then $\mu$ is called a sub-probability measure if $\mu (X)\leq 1$ .^[1]^[2]

Properties

In measure theory, the following implications hold between measures: ${\text{probability}}\implies {\text{sub-probability}}\implies {\text{finite}}\implies \sigma {\text{-finite}}$

So every probability measure is a sub-probability measure, but the converse is not true. Also every sub-probability measure is a finite measure and a σ-finite measure, but the converse is again not true.

See also

References

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