Sub-probability measure

A sub-probability measure is a measure in probability theory and measure theory that is closely related to probability measures. While probability measures always assign the value one to the underlying set, sub-probability measures assign a value smaller or equal to one to it.

Definition

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

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

References

^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 247. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
^ 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.

[Klenke247-1] Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 247. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

[Kallenberg30-2] 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.

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