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.

Definition

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

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

Properties

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.

See also

• Helly's selection theorem
• Helly–Bray theorem

References

1. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 247. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
2. Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 30. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.