sub-probability measure

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

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

In measure theory, the following implications hold between measures:

$$\backslash text\{probability\}\; \backslash implies\; \backslash text\{sub-probability\}\; \backslash implies\; \backslash text\{finite\}\; \backslash implies\; \backslash sigma\backslash text\{-finite\}$$

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.

• Helly's selection theorem
• Helly–Bray theorem

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6 |page=247}}

{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|location= Switzerland |publisher=Springer |page=30|doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3}}

{{Measure theory}}

Category:Probability theory

Category:Measures (measure theory)