Difference between revisions of "Absolutely continuous measures"
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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 | 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 | 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 $\mu (A) =0$. The absolute continuity of $\nu$ with respect to $\mu$ | + | 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 | 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 | to the following stronger statement: for any $\varepsilon>0$ there is a $\delta>0$ such that $\nu (A)<\varepsilon$ for every | ||
| − | $A$ with $\mu (A)<\delta$. | + | $A$ 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]] $\nu$ | 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 | 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 | 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 $\mu$ | + | $\nu (A) = 0$ for any $A\in\mathcal{B}$ such that $|\mu| (A)=0$, where $|\mu|$ is the [[Signed measure|total variation]] of $\mu$ |
| − | (see | + | (see for instance Theorem B, Section 31 of {{Cite|Ha}}). |
| − | The [[Radon-Nikodym theorem]] characterizes the absolute continuity of $\nu$ with respect to $\mu$ with | + | 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 | the existence of a function $f\in L^1 (\mu)$ such that $\nu = f \mu$, i.e. such that | ||
\[ | \[ | ||
\nu (A) = \int_A f\, \rd\mu \qquad \text{for every $A\in\mathcal{B}$.} | \nu (A) = \int_A f\, \rd\mu \qquad \text{for every $A\in\mathcal{B}$.} | ||
\] | \] | ||
| − | A corollary of the Radon-Nikodym, the Hahn decomposition theorem, characterizes signed measures | + | A corollary of the Radon-Nikodym, the Hahn decomposition theorem (also called Jordan decomposition theorem), characterizes signed measures |
| − | as differences of nonnegative measures. We refer to [[Signed measure]] for more on this topic. | + | 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. |
Two measures which are mutually absolutely continuous are sometimes called equivalent. | Two measures which are mutually absolutely continuous are sometimes called equivalent. | ||
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* $\nu_a$ is absolutely continuous with respect to $\mu$; | * $\nu_a$ 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 $\nu_s\perp \mu$). | * $\nu_s$ 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. | + | 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 | 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). | 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). | ||
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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 | 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]] | 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). | + | 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 $\nu\ll\mu$. | 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 $\nu\ll\mu$. | ||
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{| | {| | ||
|- | |- | ||
| − | |valign="top"|{{Ref| | + | |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|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|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}} | + | |valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}} |
|- | |- | ||
|valign="top"|{{Ref|Bi}}|| P. Billingsley, "Convergence of probability measures", Wiley (1968) {{MR|0233396}} {{ZBL|0172.21201}} | |valign="top"|{{Ref|Bi}}|| P. Billingsley, "Convergence of probability measures", Wiley (1968) {{MR|0233396}} {{ZBL|0172.21201}} | ||
| Line 60: | Line 61: | ||
|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} | |valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} | ||
|- | |- | ||
| − | |valign="top"|{{Ref| | + | |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|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 ( | + | |valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis", Macmillan (1969) {{MR|0151555}} {{ZBL|0197.03501}} |
|- | |- | ||
|} | |} | ||
Revision as of 13:42, 10 August 2012
2020 Mathematics Subject Classification: Primary: 28A33 [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-Nykodin 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]).
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 \[ \nu (A) = \int_A f\, \rd\mu \qquad \text{for every '"`UNIQ-MathJax35-QINU`"'.} \] A corollary of the Radon-Nikodym, the Hahn decomposition theorem (also called 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.
Two measures which are mutually absolutely continuous are sometimes called equivalent.
Radon-Nikdoym decomposition
If $\mu$ is a nonnegative measure on a $\sigma$-algebra $\mathcal{B}$ and $\nu$ another 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).
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 $\nu\ll\mu$.
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.
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 |
| [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 |
| [Bi] | P. Billingsley, "Convergence of probability measures", Wiley (1968) MR0233396 Zbl 0172.21201 |
| [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 |
Absolutely continuous measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely_continuous_measures&oldid=27427