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

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

Properties

In measure theory, the following implications hold between measures: $$\text{probability}⟹\text{sub-probability}⟹\text{finite}⟹\sigma \text{-finite}$$

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.

See also

• Helly's selection theorem
• Helly–Bray theorem