Motivation

Let $(X,\mathcal{F},\mu)$ be a measure space, with $\mu$ as a finite signed measure. Let $\nu$ be a measure on $(X,\mathcal{F})$ and assume $\nu\ll\mu$ (i.e. $\nu$ absolutely continuous w.r.t $\mu$). Then $\exists!f\in L^{1}(X,\mathcal{F},\mu)$ such that $$\nu(A)=\int\limits _{A}f \, d\mu$$$\forall A\in\mathcal{F}$.

Let $(X,\mathscr{M})$ be a measurable space and $\mu$ a σ-finite measure on $\mathscr{M}$. Let $\nu$ be a finite measure on $\mathscr{M}$. 1. There exists a unique pair of measures $\nu_{a},\nu_{s}$ on $\mathscr{M}$ such that: 1. $\nu=\nu_{a}+\nu_{s}$ 2. $\nu_{a}\ll \mu$ 3. $\nu_{s}\perp \mu$, (i.e. $\nu_{s},\mu$ are Mutually Singular) 2. $\exists h\in L^{1}(X,\mathscr{M},\mu)$ such that $$\nu_{a}(E)=\int\limits _{E}h \, d\mu\quad \forall E\in\mathscr{M}$$