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 $$ (X, \mathcal A) $$.

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

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

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.