Lebesgue Measure

Definition §

Given a measurable set $A$ and its indicator function $1_A$, the 1-dimensional Lebesgue measure of $A$ (denoted $|A|$ or $\text{mes}(A)$) is given by ${\int }_{-\infty }^{\infty }{1}_A(x)dx$.

Properties §

1. The intervals $(a,b)$, $(a,b]$, $[a,b)$, and $[a,b]$ all have measure $b-a$.
2. A null set $E$ has measure zero, i.e. $|E|=0$.