Measure space

A measure space is a triple $(X,\Sigma,\mu)$ where $(X,\Sigma)$ is a measurable space and $\mu$ is a measure on $\Sigma$.

Measure spaces provide the basic setting for integration, convergence theorems, and statements that hold almost everywhere rather than pointwise.

Examples:

• For any set $X$, $(X,\mathcal P(X),\mu)$ with $\mu$ the counting measure is a measure space.
• On the unit interval $[0,1]$, the triple $([0,1],\mathcal B([0,1]),\lambda)$ (Borel sets with a Lebesgue-type length measure $\lambda$) is a standard measure space used in probability and analysis.