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 [math]\displaystyle{ \mu }[/math] be a measure on the measurable space [math]\displaystyle{ (X, \mathcal A) }[/math].
Then [math]\displaystyle{ \mu }[/math] is called a sub-probability measure if [math]\displaystyle{ \mu(X) \leq 1 }[/math].[1][2]
Properties
In measure theory, the following implications hold between measures: [math]\displaystyle{ \text{probability} \implies \text{sub-probability} \implies \text{finite} \implies \sigma\text{-finite} }[/math]
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
- ↑ 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.
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