Properties of a measure having a density

Question

I am wondering what exactly are the distinguishing properties of a measure with a density (w.r.t. Leb measure) from other measures. I would like to know if there exist interesting properties of measures with density.

In particular, I know the following sufficient condition: $\liminf_{r\to0} \dfrac{\mu(B(x,r))}{Leb(B(x,r))}<\infty$ for $\mu$-almost all $x$ implies $\mu$ is absolutely continuous w.r.t. Leb.

But this is really the only sufficient condition I know. If you think about singular measures, there are notions like dimension that give an idea about how bad the measure is. But is there something similar to absolutely continuous measures? Can one compute something like dimension (or something else, maybe) that gives an idea of the absolute continuity?

I believe this falls under Classical Analysis. What are some books that talk about absolute continuity in detail so that I can learn?

Answer 1

With $\lambda$ equaling Lebesgue measure, the following has are equivalent:

• $\mu$ has a density, i.e. $\mu(A)=\int f\,d\lambda$ for all measurable $A$.
• For all $\epsilon>0$, there exists a $\delta>0$ such that for all measureable $A$, $\lambda(A)<\delta$ implies $\mu(A)<\epsilon$.
• For all measurable $A$, $\lambda(A)=0$ implies $\mu(A)=0$.

Assuming you are talking about Lebesgue measure on $\mathbb R$, there is a fourth equivalent condition:

• Letting $F(x)=\mu((0,x])-\mu((-x,0])$, then $F$ is differentiable a.e, and $F(x)=\int_0^xF'(x)\,dx$.

To simplify the definition of $F$, I am using the convention that $(0,x]=\varnothing$ when $x$ is negative.

Real Analysis by Folland gives a good exposition on all of the basics of measure theory, including these characterizations of absolute continuity. The equivalence of the first bullet with the other two is the Lebesgue-Radon-Nikodym Theorem, while the equivalence of the last with the previous is the Lebesgue Fundamental Theorem of Calculus. This book does not go into the concept of dimension of measures too much, however.