Measures

Definition (Measure)

A measure on a σ-algebra $\mathcal{A}$ on a set $X$ is mapping $\mu :\mathcal{A}\to [0,\infty ]$ such that

• $\mu (\varnothing )=0$,

• for every sequence $(A_n{)}_{n\in \mathbb{N}}$ of pairwise disjoint sets $A_n\in \mathcal{A}$

  $$\mu \left(\bigcup _{n=1}^{\infty }{A}_n\right)=\sum _{n=1}^{\infty }\mu (A_n).$$

Definition (Measure Space)

A measure space is a triple $(X,\mathcal{A},\mu )$ of a set $X$, a σ-algebra $\mathcal{A}$ on $X$ and a measure $\mu$ on $\mathcal{A}$.