Absolutely Continuous Measures

Definitions

Let $(X,{\mathcal {M}},\mu _{1})$ be a measure space. The measure $\mu _{2}:{\mathcal {M}}\rightarrow [0,\infty ]$ is said to be absolutely continuous with respect to the measure $\mu _{1}$ if we have that $\mu _{2}(T)=0$ for $T\in {\mathcal {M}}$ such that $\mu _{1}(T)=0$ (see [1]). In this case, we denote that $\mu _{2}$ is absolutely continuous with respect to $\mu _{1}$ by writing $\mu _{2}\ll \mu _{1}$ .

Examples

Recall that if $f:X\rightarrow [0,\infty ]$ is a measurable function, then the set function $\mu _{2}(T)=\int _{T}f\,d\mu _{1}$ for $T\in {\mathcal {M}}$ is a measure on $(X,{\mathcal {M}},\mu _{1})$ . Observe that if $\mu _{1}(T)=0$ , then $\mu _{2}(T)=\int _{X}f\cdot \chi _{T}\,d\mu _{1}=0$ so that $\mu _{2}\ll \mu _{1}$ .

Properties

It was previously established on a homework problem that for some nonnegative measurable $f\in L^{1}(\mu _{1})$ defined on the measure space $(X,{\mathcal {M}},\mu _{1})$ and some arbitrarily chosen $\epsilon >0$ , there exists $\delta >0$ such that $\int _{T}f\,d\mu _{1}<\epsilon$ whenever $\mu _{1}(T)<\delta$ (see [2]). The method that was used to establish this result can also be used to show that, in a finite measure space, if $\mu _{2}\ll \mu _{1}$ , then for some arbitrarily chosen $\epsilon >0$ , there exists $\delta >0$ such that $\mu _{2}(T)<\epsilon$ whenever $\mu _{1}(T)<\delta$ .