Absolutely continuous measures

2020 Mathematics Subject Classification: Primary: 28A15 [MSN][ZBL]

A concept in measure theory (see also Absolute continuity). If $\mu$ and $\nu$ are two measures on a σ-algebra $\mathcal{B}$ of subsets of $X$, we say that $\nu$ is absolutely continuous with respect to $\mu$ if $\nu (A) =0$ for any $A\in\mathcal{B}$ such that $\mu (A) =0$ (cp. with Defininition 2.11 of [Ma]). The absolute continuity of $\nu$ with respect to $\mu$ is denoted by $\nu\ll\mu$. If the measure $\nu$ is finite, i.e. $\nu (X) <\infty$, the property $\nu\ll\mu$ is equivalent to the following stronger statement: for any $\varepsilon>0$ there is a $\delta>0$ such that $\nu (A)<\varepsilon$ for every $A$ with $\mu (A)<\delta$ (this follows from the Radon-Nikodym theorem, see below, and the absolute continuity of the integral, see for instance Theorem 12.34 of [HS]).

This definition can be generalized to signed measures $\nu$ and even to vector-valued measures $\nu$. Some authors generalize it further to vector-valued $\mu$'s: in that case the absolute continuity of $\nu$ with respect to $\mu$ amounts to the requirement that $\nu (A) = 0$ for any $A\in\mathcal{B}$ such that $|\mu| (A)=0$, where $|\mu|$ is the total variation of $\mu$ (see for instance Theorem B, Section 31 of [Ha]).

Under the assumption that $\mu$ is $\sigma$-finite, the Radon-Nikodym theorem (see Theorem B of Section 31 in [Ha]) characterizes the absolute continuity of $\nu$ with respect to $\mu$ with the existence of a function $f\in L^1 (\mu)$ such that $\nu = f \mu$, i.e. such that \[ \nu (A) = \int_A f\, \rd\mu \qquad \text{for every } A\in\mathcal{B}. \] A corollary of the Radon-Nikodym theorem, the Jordan decomposition theorem, characterizes signed measures as differences of nonnegative measures (see Theorems A and B of Section 29 in [Ha]). We refer to Signed measure for more on this topic. See also Hahn decomposition.

Two measures which are mutually absolutely continuous are sometimes called equivalent.

Radon-Nikodym decomposition

If $\mu$ is a $\sigma$-finite nonnegative measure on a $\sigma$-algebra $\mathcal{B}$ and $\nu$ another $\sigma$-finite nonnegative measure on the same $\sigma$-algebra (which might be a signed measure, or even taking values in a finite-dimensional vector space), then $\nu$ can be decomposed in a unique way as $\nu=\nu_a+\nu_s$ where

• $\nu_a$ is absolutely continuous with respect to $\mu$;
• $\nu_s$ is singular with respect to $\mu$, i.e. there is a set $A$ of $\mu$-measure zero such that $\nu_s (X\setminus A)=0$ (this property is often denoted by $\nu_s\perp \mu$).

This decomposition is called Radon-Nikodym decomposition by some authors and Lebesgue decomposition by some other (see Theorem C of Section 32 in [Ha]). The same decomposition holds even if $\nu$ is a signed measure or, more generally, a vector-valued measure. In these cases the property $\nu_s (X\setminus A)=0$ is substituted by $\left|\nu_s\right| (X\setminus A)=0$, where $\left|\nu_s\right|$ denotes the total variation measure of $\nu_s$ (we refer to Signed measure for the relevant definition).

Some authors use the name "Differentiation of measures" for the decomposition above and the density $f$ is sometimes denoted by $\frac{d\nu}{d\mu}$ or $D_\mu \nu$ (see for instance Section 32 of [Ha]). Other authors use the term "Differentiation of measures" for a theorem, due to Besicovitch, which, for Radon measures in the Euclidean space, characterizes $f(x)$ as the limit of a suitable quantity, see Differentiation of measures for the precise statement.

Comments

A set of non-zero measure that has no subsets of smaller, but still positive, measure is called an atom of the measure. When considering the $\sigma$-algebra $\mathcal{B}$ of Borel sets in the euclidean space and the Lebesgue measure $\lambda$ as reference measure, it is a common mistake to claim that the singular part of a second measure $\nu$ must be concentrated on points which are atoms. A singular measure may be atomless, as is shown by the measure concentrated on the standard Cantor set which puts zero on each gap of the set and $2^{-n}$ on the intersection of the set with the interval of generation $n$ (such measure is also the distributional derivative of the Cantor ternary function or devil staircase, (see Problem 46 in Chapter 2 of [Ro]).

When some canonical measure $\mu$ is fixed, (as the Lebesgue measure on $\mathbb R^n$ or its subsets or, more generally, the Haar measure on a topological group), one says that $\nu$ is absolutely continuous meaning that $\nu\ll\mu$.

References

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