# Absolute Continuity (Part Two)

/In last week's post,** **we saw how absolute continuity - of a function *and* of a measure - is key in any discussion of Lebesgue's Fundamental Theorem of Calculus. In particular, these two types of absolute continuity appear in the proof that

*if $f:[a,b]\to[0,\infty]$ is an integrable function and $F(x)=\int_a^xf\;d\mu$, then $F$ is absolutely continuous. *

The bulk of the proof (and where we see aboslute continuity in action) stems from the following

*Lemma*: For any (Lebesgue) measurable set $A$ and for all $\epsilon>0$ there is a $\delta>0$ such that $\mu(A)<\delta$ implies $\int_Af\;d\mu< \epsilon$.

**Step 1**

**Claim**: Let $\nu$ and $\mu$ be positive measures, with $\nu$ finite*, on a measure space $(X,\Sigma)$. If $\nu$ is absolutely continuous with respect to $\mu$ (i.e. $\nu\ll\mu$), then for all $\epsilon>0$ there exists $\delta>0$ such that $\mu(A)< \delta$ implies $\nu(A)< \epsilon$ for all measurable sets $A$.

*Proof:* Suppose $\nu \ll \mu$, i.e. $\nu(A)=0$ whenever $\mu(A)=0$ for any measurable set $A$. But suppose to the contrary that there exists an $\epsilon>0$ such that for all $n\in\mathbb{N}$,
$$\mu(A_n)<\frac{1}{2^n} \quad \text{yet} \quad \nu(A_n)\geq \epsilon \quad \text{for each $n$.}$$
For each $k\in\mathbb{N}$, define $$F_k=\bigcup_{n=k}^\infty A_n$$ and observe that $F_1\supset F_2\supset F_3\supset \cdots$. Thus by continuity from above**
\begin{align*}
\mu\left( \bigcap_{k=1}^\infty F_k \right) &= \lim_{k\to\infty}\mu (F_k)\\
&= \lim_{k\to\infty}\mu \left( \bigcup_{n=k}^\infty A_n \right)\\
&\leq \lim_{k\to\infty} \sum_{n=k}^\infty \mu(A_n)\\
&=\lim_{k\to\infty} \frac{1}{2^{k-1}}\\
&=0.
\end{align*}
On the other hand, since $\nu(A_n)\geq \epsilon$ for all $n$, it follows that for every $k$,
$$\nu(F_k)=\nu\left( \bigcup_{n=k}^\infty A_n \right)\geq \epsilon$$
and so again by continuity from above,
$$\nu\left( \bigcap_{k=1}^\infty F_k\right)=\lim_{k\to\infty}\nu(F_k)\geq \epsilon.$$
Thus, if we let $A=\bigcap_{k=1}^\infty \bigcup_{n=k}^\infty A_n$ we see that $\mu(A)=0$ yet $\nu(A)\geq \epsilon$, contradicting the fact that $\nu \ll \mu$.

## Step 2

**Claim**: Suppose $\nu$ and $\mu$ are positive measures on a measure space $(X,\Sigma)$. For any measurable set $A$ and integrable function $f:X\to[0,\infty]$, define $$\nu(A)=\int_Af\;d\mu.$$ Then $\nu$ is a measure.

*Proof:* To verify that $\nu$ is a measure, we have to check three things:

- That $\nu(A)\geq 0$ for all measurable sets $A$ is clear since we're assuming $f$ itself is a positve function.***
- Since $\mu$ is a measure, we know $\mu(\varnothing)=0$. Thus $\nu(\varnothing)=\int_{\varnothing}f\;d\mu=0$ too.
*(In other words, $\nu \ll \mu$.)* - Suppose $\{A_n\}$ is a countable collection of pairwise disjoint measurable sets. Then \begin{align*} \nu\left(\bigcup_{n=1}^\infty A_n\right) &= \int_{\cup_n A_n}f\;d\mu\\ &=\sum_{n=1}^\infty\int_{a_n} f\;d\mu\\ &=\sum_{n=1}^\infty\nu(A_n) \end{align*} which shows that $\nu$ is countably additive.

## Step 3

**Claim**: The Lemma is proved!

*Proof:* Let $\nu$ and $\mu$ be defined as in Step 2 and observe that $\nu$ is absolutely continuous with respect to $\mu$. Thus by Step 1, for all $\epsilon>0$ there is a $\delta>0$ such that $\mu(A) < \delta$ implies $\nu(A)=\int_Af\;d\mu < \epsilon$ for all measurable sets $A$. This is precisely the statement of the Lemma.

*Footnotes:*

* [Edited 3/2/16] We need $\nu$ to be a finite measure! Many thanks to a reader for pointing this out in the comments.

** Notice that $\mu(F_1)<\infty$ since \begin{align*} \mu(F_1)&=\mu\left(\bigcup_{n=1}^\infty A_n \right)\\ &\leq \sum_{n=1}^\infty \mu(A_n)\\ &=1. \end{align*}

*** If $f$ is allowed to take on negative values, the math still holds, but we would need to work with a *signed* measure instead.