tags:definition |analysis

Lebesgue Measure

Statements

Definition. Let $(X,\mathfrak{B}$) be a measurable space, where $\mathfrak{B}$ is a Borel sigma algebra. The measure $\mu :X\to {\mathbb{R}}_{\ge 0}$ is said to be a Lebesgue measure if

1. Every open set and closed set belong to $\mathfrak{B}$.
2. If $A\in \mathfrak{B}$ and $\mu (A)=0$, then every subset of $A$ belongs to $\mathfrak{B}$ also has also measure zero.
3. if $\{A_k{\}}_{k\in \mathbb{N}}\subset \mathfrak{B}$ then also $\cup A_k$ and $\cap A_k$ belong to $\mathfrak{B}$.

References

@Sal08, Theorem B.1