View source for Absolutely continuous measures

{{MSC|28A15}}

[[Category:Classical measure theory]]

{{TEX|done}}

A concept in measure theory (see also [[Absolute continuity]]). If $\mu$ and $\nu$ are two measures on a [[Algebra of sets|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 {{Cite|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-Nykodin theorem, see below, and the absolute continuity of the integral, see for instance
Theorem 12.34 of {{Cite|HS}}).

This definition can be generalized to [[Signed measure|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 [[Signed measure|total variation]] of $\mu$
(see for instance Theorem B, Section 31 of {{Cite|Ha}}).

Under the assumption that $\mu$ is $\sigma$-finite, the [[Radon-Nikodym theorem]] (see Theorem B of Section 31 in {{Cite|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 (of a signed measure)|Jordan decomposition theorem]], characterizes signed measures
as differences of nonnegative measures (see Theorems A and B of Section 29 in {{Cite|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 measures|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 {{Cite|Ha}}).
The same decomposition holds even if $\nu$ is a [[Signed measure|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 {{Cite|Ha}}). Other authors use the term "Differentiation of measures" for a theorem, due to Besicovitch, which, for [[Radon measure|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 set|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|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 [[Generalized derivative|distributional derivative]]
of the Cantor ternary function or devil staircase, (see Problem 46 in Chapter 2 of {{Cite|Ro}}). 

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

For Radon measures in the Euclidean space, both the Radon-Nikodym theorem and the Radon-Nikodym decomposition can be proved using the Besicovitch [[Differentiation of measures|differentiation of measures]].


====References====
{|
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|valign="top"|{{Ref|AFP}}||     L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations    and  free discontinuity problems". Oxford Mathematical Monographs. The     Clarendon Press, Oxford University Press, New York, 2000.     {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|valign="top"|{{Ref|Bi}}||     P. Billingsley, "Convergence  of probability measures", Wiley (1968)     {{MR|0233396}}  {{ZBL|0172.21201}}
|-
|valign="top"|{{Ref|Bo}}||     N. Bourbaki, "Elements of mathematics. Integration", Addison-Wesley     (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}}     {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}}     {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}}     {{ZBL|1095.28001}} {{ZBL|0156.06001}}
|-
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|-
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
|-
|valign="top"|{{Ref|HS}}||   E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis", Springer   (1965) {{MR|0188387}} {{ZBL|0137.03202}}
|-|valign="top"|{{Ref|Ma}}||     P. Mattila, "Geometry of sets and measures in euclidean spaces".     Cambridge Studies in Advanced Mathematics, 44. Cambridge University     Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
|-
|valign="top"|{{Ref|Ro}}|| H.L. Royden,   "Real analysis", Macmillan  (1969) {{MR|0151555}} {{ZBL|0197.03501}}
|-
|}