Definition (Signed Measure)

Let $(X,\mathcal{F})$ be a measurable space. A signed measure $\mu$ on $(X,\mathcal{F})$ is a mapping such that

1. $\mu:X\to (-\infty,\infty)$ such that $\mu(\emptyset)=0$
2. $\forall(A_{n})_{n\in\mathbb{N}}\subset\mathcal{F}$ s.t. $A_{i}\cap A_{j}=\emptyset$ (i.e. pairwise disjoint) we have $$\mu\left( \bigcup_{n=1}^{\infty}A_{n} \right)=\sum_{n=1}^{\infty}\mu(A_{n})$$

Definition (Positive Signed Measure)

Let $(X,\mathcal{F})$ be a measurable space, let $\mu$ be a signed measure on $(X,\mathcal{F})$. We define the positive signed measure to be $$\mu^{+}(A)=\sup_{C\subset A,C\in\mathcal{F}}\mu(C)$$

Definition (Negative Signed Measure)

Let $(X,\mathcal{F})$ be a measurable space, let $\mu$ be a signed measure on $(X,\mathcal{F})$. We define the negative signed measure to be $$\mu^{-}(A)=-\inf_{C\subset A,C\in\mathcal{F}}\mu(C)$$