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Suppose that on the [[Measurable space|measurable space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300301.png" /> there are given two measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300303.png" /> (cf. also [[Measure|Measure]]). One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300304.png" /> is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300305.png" /> (denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300306.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300307.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300308.png" /> for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a1300309.png" />. One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003010.png" /> dominates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003011.png" />. If the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003012.png" /> is finite (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003013.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003014.png" /> if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003015.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003017.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003018.png" />.
+
{{MSC|28A15}}
  
The [[Radon–Nikodým theorem|Radon–Nikodým theorem]] says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003020.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003021.png" />-finite measures and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003022.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003023.png" />-integrable non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003024.png" /> (a density, cf. also [[Integrable function|Integrable function]]), called the Radon–Nikodým derivative, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003025.png" />. Two such densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003027.png" /> may differ only on a null set (see [[Measure|Measure]]), i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003028.png" />. An example of a density (with respect to the [[Lebesgue measure|Lebesgue measure]] on the interval, i.e. the length) is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003030.png" /> is the sequence of all rational numbers in this interval.
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[[Category:Classical measure theory]]
  
The measure is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003031.png" />-finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003032.png" /> is the union of a countable family of sets with finite measure.
+
{{TEX|done}}
  
Given a reference measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003034.png" />, any measure may be decomposed into a sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003036.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003038.png" />, i.e. an absolutely continuous and a singular part. This is called the Lebesgue decomposition.
+
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-Nikodym theorem, see below, and the absolute continuity of the integral, see for instance
 +
Theorem 12.34 of {{Cite|HS}}).
  
A set of non-zero measure that has no subsets of smaller, but still positive, measure is called an atom of the measure. It is a common mistake to claim that the singular part of a measure 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003039.png" /> on the intersection of the set with the interval of generation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003040.png" />.
+
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}}).
  
When some canonical measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003041.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003042.png" /> is fixed (as the [[Lebesgue measure|Lebesgue measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003043.png" /> or its subsets or, more generally, the [[Haar measure|Haar measure]] on a [[Topological group|topological group]]), one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003044.png" /> is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003045.png" />, meaning that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130030/a13003046.png" />.
+
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 called equivalent.
+
Two measures which are mutually absolutely continuous are sometimes called equivalent.
  
See also [[Absolute continuity|Absolute continuity]].
+
====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|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 $\nu\ll\mu$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.L. Royden, [[Royden, "Real analysis"|"Real analysis"]], Macmillan  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"E. Hewitt,  K. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR></table>
+
{|
 +
|-
 +
|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}}
 +
|-
 +
|valign="top"|{{Ref|DS}}||    N. Dunford, J.T. Schwartz, "Linear operators. General theory",   '''1''', Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
 +
|-
 +
|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}}
 +
|-
 +
|}

Latest revision as of 07:56, 15 December 2016

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

[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. MR1857292Zbl 0957.49001
[Bi] P. Billingsley, "Convergence of probability measures", Wiley (1968) MR0233396 Zbl 0172.21201
[Bo] N. Bourbaki, "Elements of mathematics. Integration", Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958) MR0117523 Zbl 0635.47001
[Ha] P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[HS] E. Hewitt, K.R. Stromberg, "Real and abstract analysis", Springer (1965) MR0188387 Zbl 0137.03202
[Ro] H.L. Royden, "Real analysis", Macmillan (1969) MR0151555 Zbl 0197.03501
How to Cite This Entry:
Absolutely continuous measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely_continuous_measures&oldid=25515
This article was adapted from an original article by T. Nowicki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article