On Dominating Measures and Absolute Continuity

# On Dominating Measures and Absolute Continuity Motivation: I see this often in theory sections, e.g. when $\nu$ dominates $\mu$ ($\nu \gg \mu$)... or equivalently $\mu$ is absolutely continuous with respect to $\nu$... Sadly, I do not have an intuition about why this is important or what I should know or feel when I see this. TLDR conclusion: The two notions are equivalent. That $\nu \gg \mu$ allows us to use Radon-Nikodym to define a unique density $f$ of $\mu$ with respect to $\nu$. We call $f$ the Radon-Nikodym derivative. **Most commonly, absolute continuity is a criterion that lets us claim that the measure of interest admits a density with respect to the Lebesgue measure.** ## Absolute Continuity (of a measure) From [Wiki][ac wikipedia]: > If $\mu$ and $\nu$ are two measures on the same measurable space $(X,\mathcal{A})$, $\mu$ is said to be **absolutely continuous** with respect to $\nu$ if $\mu(A) = 0$ for every set $A$ for which $\nu(A) = 0$. This is written as "$\mu \ll \nu$". That is: $$\mu \ll \nu \qquad \iff \qquad \forall A\in {\mathcal {A}}\quad (\nu (A)=0\ \Rightarrow \ \mu (A)=0).$$ When $\mu \ll \nu$, we say $\nu$ is **dominating** $\mu$. If you're wondering about the notion of continuity a la\' $\epsilon-\delta$, then you're thinking about the absolute continuity of a *function*. Indeed, it is a theorem that $\mu$ is absolutely continuous with respect to $\nu$ if and only if for all $A \in \mathcal{A}$ and for all $\epsilon > 0$, there exists $\delta >0$ such that $\nu(A)<\delta \Rightarrow \mu(A)<\epsilon$. Lebesgue's Fundamental Theorem of Calculus relates the two notions of absolute continuity and the equivalence is explained nicely in a [Math3ma](https://www.math3ma.com) blog post: [part 1][ac math3ma 1] and [part 2][ac math3ma 2]. This is not necessary for our discussion. Let's move on. Aside: Also, there is some wonderful content at Math3ma about functions and measures under the ['analysis'](https://www.math3ma.com/categories/analysis) tag. Do yourself a favour and check it out. ## The Radon-Nikodym Derivative Again from the same [Wiki][ac wikipedia]: > The Radon–Nikodym theorem states that if $\mu$ is absolutely continuous with respect to $\nu$, and both measures are $\sigma$-finite, then $\mu$ has a density, or 'Radon-Nikodym derivative', with respect to $\nu$, which means that there exists a $\nu$-measurable function $f$ taking values in $[0, \infty)$, denoted by $f = \frac{d\mu}{d\nu}$, such that for any $\nu$-measurable set $A$ we have $$\mu (A)=\int _{A}f\,d\nu.$$ Recall that a measure (also called a distribution) defines uniquely a distribution function (cdf). However, the existence of a density (pdf) is not guaranteed. For example, the distribution of discrete random variables do not admit densities. A continuous example is the [Devil's Staircase][Cantor distribution]. However, $\nu \gg \mu$ tells us that by Radon-Nikodym there exists *the* density $f$ of the measure $\mu$ 'in terms of the measure $\nu$'. The $\sigma$-finite property tells us that the measure is a probability measure. See the [Wiki][sigma-finite] for a proof. ## Personal Note Though I'm no expert in measure-theoretic analysis, these notions are in fact not terribly difficult to grasp. In fact, the bulk of it is in one Wikipedia page, which I have visited probably many times. Thing is, I do not even remember the conditions for Radon-Nikodym to begin with. This is because when I first learnt it I did not arrive at the conclusion that they were crucial. So, it was not until I read this [Stack Exchange post][ac stackexchange] that I realise the point of absolute continuity is to access Radon-Nikodym to define a density. Oh well, better late than never. Though, now I must wonder how many more things I could know but do not. ## References * [Wikipedia: Absolute Continuity][ac wikipedia] * [Math3ma: Absolute Continuity (Part One)][ac math3ma 1] * [Math3ma: Absolute Continuity (Part Two)][ac math3ma 2] * [Wikipedia: Cantor Distribution][cantor distribution] * [Wikipedia: $\sigma$-finite measure][sigma-finite] * [Stack Exchange: Intuition on Dominating Measures and Absolute Continuity][ac stackexchange] [ac wikipedia]: <https://en.wikipedia.org/wiki/Absolute_continuity> [ac math3ma 1]: <https://www.math3ma.com/blog/absolute-continuity-part-one> [ac math3ma 2]: <https://www.math3ma.com/blog/absolute-continuity-part-two> [cantor distribution]: <https://en.wikipedia.org/wiki/Cantor_distribution> [sigma-finite]: <https://en.wikipedia.org/wiki/%CE%A3-finite_measure#Equivalence_to_a_probability_measure> [ac stackexchange]: <https://math.stackexchange.com/questions/3324089/intuition-on-dominating-measures-and-absolute-continuity>