Sub-probability measure explained

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

\mu be a measure on the measurable space (X, l A) .

Then

\mu is called a sub-probability measure if \mu(X) \leq 1 .

Properties

In measure theory, the following implications hold between measures:$$\backslash text\; \backslash implies\; \backslash text\; \backslash implies\; \backslash text\; \backslash implies\; \backslash sigma\backslash text$$

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] [2]}

Notes and References

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