Sub-probability measure

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

Then ${\displaystyle \mu }$ is called a sub-probability measure if ${\displaystyle \mu (X)\leq 1}$.¹²

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

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.