Why are absolutely continuous measures called that way?

Question

Let $X\subseteq \Bbb{R}^n$ and $\mathcal{B}(X)$ the borelian set for $X$. Let $\mu$ and $\nu$ be two measures on $(X,\mathcal{B})$. We say that $\mu$ is absolutely continuous with respecto to $\nu$ (denoted by $\mu\ll \nu$) if $\forall A\in\mathcal {B}(X),\; \mu(A)=0\implies \nu(A)=0$.

Generally, continuity refers to some sort of smoothness of a function: small variations on domain gives an small variation on co-domain. I don't see how this definition fits this category which leads me to wonder, why are absolutely continuous measures called that way? I'm looking for maybe an historical answer (the reason why it started being called that way) or an answer that appeals to the definition itself (something on the definition which makes it reasonable to be called that way).

Answer 1

The conditions $$ \forall A \in \mathcal B(X): \quad ( \mu(A) = 0 \;\Rightarrow\; \nu(A) = 0 ) $$ and $$ \forall \varepsilon > 0 : \exists \delta > 0 : \forall A \in \mathcal B(X): \quad ( \mu(A) \le \delta \;\Rightarrow\; \nu(A) \le \varepsilon ) $$ in case that $\mu$ is $\sigma$-finite and $\nu$ is finite. The direction "$\Leftarrow$" is clear, while "$\Rightarrow$" follows from the theorem of Radon-Nikodým and the absolute continuity of the Lebesgue integral.

Finally, we note that the second condition is very similar to the absolute continuity of a function. Indeed, a function $F \colon [a,b] \to \mathbb R$ is called absolutely continuous, if \begin{equation*} \sum_{i = 1}^n (y_i - x_i) \le \delta \quad\Rightarrow\quad \sum_{i = 1}^n | F(y_i) - F(x_i) | \le \varepsilon. \end{equation*} holds for arbitrary disjoint subintervalls $(x_i,y_i) \subset [a,b]$, $x_i < y_i$, $i = 1,\ldots, n$, $n \in \mathbb N$. Now, let $\mu$ be the Lebesgue measure on $\mathbb R$ and $\nu$ the Lebesgue-Stieltjes measure defined via $F$. Then, the absolute continuity of $F$ is equivalent to $$ \forall \varepsilon > 0 : \exists \delta > 0 : \forall A \in \mathcal F: \quad ( \mu(A) \le \delta \;\Rightarrow\; \nu(A) \le \varepsilon ), $$ where $\mathcal F$ contains all unions of finitely many intervals.

Answer 2

Continuity actually doesn't have anything to do with smoothness, only with the existence of a limit at a point.

A function is continuous at $x=a$ iff:

$$\lim_{x\to a} f(x) = f(a)$$

I agree that limits of functions are defined such that we can always find an interval in $x$ that corresponds to a particular interval around the limit $L$ of $f(x)$ (the usual $\delta-\epsilon$ definition).

Smoothness is related to the differentiability of a function, so the continuity of its derivative.

A pretty standard counterexample of this is the Weierstrass Function, which is hardly smooth but is, nonetheless, continuous.

The concept of absolutely continuous above is related to the $\delta-\epsilon$ definition but here we are only dealing with general sets and their measure. These two notions are related, but not the same (see below).

An example of a function that is continuous, but not absolutely continuous, is the Cantor Function -- it is not the integral of its pointwise derivative, which is 0 almost everywhere, except on an uncountable set of measure 0.

As for why "absolutely" this may be to compare it to "absolutely convergent" infinite series, which is the type of convergence needed work with infinite series in a way analogous to finite series. In a similar way, absolute convergence is stronger than pointwise or uniform convergence.

This seems like the strongest historical link -- in that absolutely continuous functions behave like "nice" functions from calculaus -- they are the integrals of their derivatives (unlike the singular functions, of which the Cantor Function is one example)

Answer 3

Comment. I guess you would need to consult G. Vitali. (1905, coined the term "absolutely continuous function"; later "absolutely continuous measure" was modeled on that.)

From Mathwords:

ABSOLUTE CONTINUITY. The concept was introduced in 1884 by E. Harnack "Die allgemeinen Sätze über den Zusammenhang der Functionen einer reellen Variabelen mit ihren Ableitungen. II. Theil." Math. Ann. 21, (1884), 217-252. The term was introduced in 1905 by G. Vitali. In the meantime several mathematicians had used the concept. See T. Hawkins Lebesgue's Theory of Integration.