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.

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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.