# Absolutely Continuous Measures

## Definitions

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

## Examples

Recall that if ${\displaystyle f:X\rightarrow [0,\infty ]}$ is a measurable function, then the set function ${\displaystyle \mu _{2}(T)=\int _{T}f\,d\mu _{1}}$ for ${\displaystyle T\in {\mathcal {M}}}$ is a measure on ${\displaystyle (X,{\mathcal {M}},\mu _{1})}$. Observe that if ${\displaystyle \mu _{1}(T)=0}$, then ${\displaystyle \mu _{2}(T)=\int _{X}f\cdot \mathbb {1} _{T}\,d\mu _{1}=0}$ so that ${\displaystyle \mu _{2}\ll \mu _{1}}$ (see [3] for further details on this example and others).

## Properties

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

In particular, we proceed by contradiction and suppose there exists ${\displaystyle \epsilon >0}$ so that for any ${\displaystyle \delta >0}$ and ${\displaystyle \mu _{1}(T)<\delta }$, we have ${\displaystyle \mu _{2}(T)\geq \epsilon }$. Now, define a sequence of sets ${\displaystyle \{T_{n}\}_{n\in \mathbb {N} }\subseteq {\mathcal {M}}}$ such that ${\displaystyle \mu _{1}(T_{n})<{\frac {\epsilon }{2^{n}}}}$ and denote ${\displaystyle T=\limsup {T_{n}}=\cap _{n=1}^{\infty }G_{n}\in {\mathcal {M}}}$ where ${\displaystyle G_{n}=\cup _{k=n}^{\infty }T_{k}}$. We have from countable subadditivity that ${\displaystyle \mu _{1}(G_{n})\leq \sum _{k=n}^{\infty }\mu _{1}(T_{k})<\sum _{k=n}^{\infty }{\frac {\epsilon }{2^{k}}}={\frac {\epsilon }{2^{n+1}}}\,\forall n\in \mathbb {N} }$. We have from monotonicity that ${\displaystyle \mu _{2}(G_{n})\geq \epsilon \,\,\forall n\in \mathbb {N} }$. The monotonicity of the measure ${\displaystyle \mu _{1}}$ implies that ${\displaystyle \mu _{1}(\cap _{n=1}^{\infty }G_{n})=\mu _{1}(\limsup {T_{n}})=\mu _{1}(T)=0<\delta }$. Applying continuity from above to ${\displaystyle \mu _{2}}$, we also have ${\displaystyle \mu _{2}(T)\geq \epsilon }$. However, this contradicts the definition of ${\displaystyle \mu _{2}\ll \mu _{1}}$.

In fact, the converse to the above result also holds (see [3]). Namely, if we have ${\displaystyle \forall \epsilon >0}$ that there exists ${\displaystyle \delta >0}$ so that ${\displaystyle \mu _{1}(T)<\delta \implies \mu _{2}(T)<\epsilon }$, then ${\displaystyle \mu _{2}\ll \mu _{1}}$. Suppose that ${\displaystyle \mu _{1}(T)=0}$ for some ${\displaystyle T\in {\mathcal {M}}}$. Then, for any such ${\displaystyle \epsilon >0}$ as in the preceding claim, we have ${\displaystyle \mu _{1}(T)=0<\delta \implies \mu _{2}(T)<\epsilon }$. Since ${\displaystyle \epsilon >0}$ can be taken to be arbitrarily small, we have that ${\displaystyle \mu _{2}(T)=0}$, as required for the measure ${\displaystyle \mu _{2}}$ to be absolutely continuous with respect to ${\displaystyle \mu _{1}}$.

## References

[1]: Taylor, M. "Measure Theory and Integration". 50-51.

[2]: Craig, K. "Math 201a: Homework 8". Fall 2020. Refer to question 2.

[3]: Rana, I. K. "Introduction to Measure and Integration". Second Edition. 311-313.