Signed Measures

Definition (Signed Measure)

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

• $\mu (\varnothing )=0$,
• either there is no $A\in \mathcal{A}$ with $\mu (A)=-\infty$ or there is no $A\in \mathcal{A}$ with $\mu (A)=\infty$,

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

  $$\mu \left(\bigcup _{n=1}^{\infty }{A}_n\right)=\sum _{n=0}^{\infty }\mu (A_n).$$

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 $\mathcal{A}$.