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

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Let   be a measure on the measurable space  .

Then   is called a sub-probability measure if  .[1][2]

Properties

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In measure theory, the following implications hold between measures:  

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

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References

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  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.