Difference between revisions of "Absolutely continuous measures"
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− | + | {{MSC|28A33}} | |
− | + | [[Category:Classical measure theory]] | |
− | + | {{TEX|done}} | |
− | + | A concept in measure theory (see also [[Absolute continuity]]). If $\mu$ and $\nu$ are two measures on a $\sigma$-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 $\mu (A) =0$. The absolute continuity of $\nu$ with respect to $\mu$ | ||
+ | is denoted by $\nu<<\mu$. If the measure $\nu$ is finite, i.e. $\nu (X) <\infty$, the property $\nu<<\mu$ 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 definition can be generalized to [[Signed measure|signed measures]] $\nu$ | |
+ | and even to vector-valued measure $\nu$. 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 [[Signed measure]] for the relevant definition). | ||
− | + | The [[Radon-Nikodym theorem]] 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, the Hahn decomposition theorem, characterizes signed measures | ||
+ | as differences of nonnegative measures. We refer to [[Signed measure]] for more on this topic. | ||
− | Two measures which are mutually absolutely continuous are called equivalent. | + | 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 absolutey 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 decompoition by some authors and Lebesgue decomposition by some other. | ||
+ | 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 $|\nu_s| (X\setminus A)=0$, where $|\nu_s|$ | ||
+ | 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 Borel $\sigma$-algebra $\mathcal{B}$ in the euclidean space and the 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). | ||
+ | |||
+ | 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 $\nu<<\mu$. | ||
− | |||
====References==== | ====References==== | ||
− | + | {| | |
+ | |- | ||
+ | |valign="top"|{{Ref|AmFuPa}}|| 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|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Bi}}|| P. Billingsley, "Convergence of probability measures" , Wiley (1968) {{MR|0233396}} {{ZBL|0172.21201}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|He}}|| 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 (1968) | ||
+ | |- | ||
+ | |} |
Revision as of 15:09, 30 July 2012
2010 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 $\sigma$-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 $\mu (A) =0$. The absolute continuity of $\nu$ with respect to $\mu$ is denoted by $\nu<<\mu$. If the measure $\nu$ is finite, i.e. $\nu (X) <\infty$, the property $\nu<<\mu$ 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 definition can be generalized to signed measures $\nu$ and even to vector-valued measure $\nu$. 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 Signed measure for the relevant definition).
The Radon-Nikodym theorem 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 \[ \nu (A) = \int_A f\rd\mu \qquad \text{for every '"`UNIQ-MathJax36-QINU`"'.} \] A corollary of the Radon-Nikodym, the Hahn decomposition theorem, characterizes signed measures as differences of nonnegative measures. 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 absolutey 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 decompoition by some authors and Lebesgue decomposition by some other. 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 $|\nu_s| (X\setminus A)=0$, where $|\nu_s|$ 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 Borel $\sigma$-algebra $\mathcal{B}$ in the euclidean space and the 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).
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 $\nu<<\mu$.
References
[AmFuPa] | 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 |
[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 |
[He] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202 |
[Ro] | H.L. Royden, "Real analysis" , Macmillan (1968) |
Absolutely continuous measures. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Absolutely_continuous_measures&oldid=27253