Absolutely constant measure

The concept of absolutely continuous measure relates the zero sets of different measures in measure theory . Absolutely continuous measures are closely related to the absolutely continuous functions of analysis and the absolutely continuous distributions of probability theory .

definition

It is a measuring room and and two ( signed , complex or positive) dimensions on . ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle X}$

The measure is called absolutely continuous with respect to (also -continuous ), in characters , if every -null set is also a -null set. ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \nu \ll \mu }$ ${\displaystyle \mu }$ ${\displaystyle \nu }$

For any measurable amount therefore follows from well . Conversely, one then says that measure dominates . Through one is quasi-ordering on the set of dimensions to explain. ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle \mu (A)=0}$ ${\displaystyle \nu (A)=0}$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle \ll }$ ${\displaystyle X}$

Examples

The zero measure, which assigns the measure to every quantity , is naturally dominated by every measure. ${\displaystyle 0}$

Let the counting measure on the natural numbers , more precisely on the measuring room . Then each measure on respect. Absolutely continuous, because the only -null is the empty set . ${\displaystyle \mu }$ ${\displaystyle (\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ))}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \emptyset }$

The probability measure of the standard normal distribution has a probability density with respect to the Lebesgue measure , because for every Lebesgue measurable set applies ${\displaystyle P}$ ${\displaystyle \lambda }$ ${\displaystyle A\subseteq \mathbb {R} }$

${\displaystyle P(A)=\int _{A}\,{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}\ \mathrm {d} \lambda (x)}$.

It follows from this that every Lebesgue null set of is also assigned the probability , that is . For example is . ${\displaystyle P}$ ${\displaystyle 0}$ ${\displaystyle P\ll \lambda }$ ${\displaystyle P(\mathbb {Q} )=0}$

The last example can be generalized. Suppose a measure can be represented by a density function with respect to another measure , so it applies to every set from the σ-algebra . Then is , because the integral over a null set is always . ${\displaystyle \nu }$ ${\displaystyle f_{\nu }\colon X\to \mathbb {R} }$ ${\displaystyle \mu }$ ${\displaystyle \textstyle \nu (A)=\int _{A}f_{\nu }\ \mathrm {d} \mu }$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \nu \ll \mu }$ ${\displaystyle 0}$

The reverse is generally not true . Thus the Lebesgue measure with regard to the counting measure is absolutely continuous, but has no density. For certain special cases, however, a reversal can be specified (see below ). ${\displaystyle \mathbb {R} }$

Characterizations

In certain cases, properties of dimensions can be specified that are equivalent to the definition above. Let be a positive measure and a finite or complex measure on the same measurement space, so in particular be . The following sentence then applies: ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle |\nu (X)|<\infty }$

The measure is exactly then absolutely continuous wrt. When each one is, such that for all with valid . ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle \mu (A)<\delta }$ ${\displaystyle |\nu (A)|<\varepsilon }$

If on the other hand , the first part no longer implies the second. ${\displaystyle |\nu (X)|=\infty }$

Denote again the Lebesgue measure on the real line and another measure . The distribution function of is defined as ${\displaystyle \lambda }$ ${\displaystyle \nu }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle F_{\nu }}$ ${\displaystyle \nu }$ ${\displaystyle F_{\nu }\colon \mathbb {R} \to [0,\infty ];\,x\mapsto \nu ((-\infty ,x]).}$

The measure is exactly then absolutely continuous wrt. When any limitation of a finite interval an absolutely continuous function on is. ${\displaystyle \nu }$ ${\displaystyle \lambda }$ ${\displaystyle F_{\nu }}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle I}$

The first characterization shows that absolute continuity is actually a concept of continuity for dimensions. The second characterization motivates the designation.

Equivalence of dimensions

Since there is a quasi-order, one can get through ${\displaystyle \ll }$

${\displaystyle \nu \sim \mu :\Longleftrightarrow \nu \ll \mu \land \mu \ll \nu }$

define an equivalence relation on the set of all measures on . For equivalent dimensions, the zero quantities match exactly. The equivalence classes are semi-ordered by . ${\displaystyle X}$ ${\displaystyle \ll }$

This equivalence explains many useful properties, for example of σ-finite measures, because:

If σ-finite, then it is equivalent to a finite measure; even if . ${\displaystyle \mu }$ ${\displaystyle \mu (X)=\infty }$

In addition, there is an integrated function , so that for everyone . The equivalent finite measure is then given by, i.e. H. is the density of . If this is not the zero dimension, you can choose that . The measure is then even equivalent to a probability measure. ${\displaystyle \mu }$ ${\displaystyle w\in L^{1}(\mu )}$ ${\displaystyle 0<w(x)<1}$ ${\displaystyle x\in X}$ ${\displaystyle \nu }$ ${\displaystyle \textstyle \nu (A)=\int _{A}w\ \mathrm {d} \mu }$ ${\displaystyle w}$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle \mu \not \equiv 0}$ ${\displaystyle w}$ ${\displaystyle \textstyle \int _{X}w\ \mathrm {d} \mu =1}$ ${\displaystyle \mu }$

In fact, the above sentence can be reinforced as follows:

If s-finite and , then it is equivalent to a probability measure. ${\displaystyle \mu }$ ${\displaystyle \mu \not \equiv 0}$

This is a real generalization, since σ-finite measures are always s-finite, but not vice versa.

σ-finite measures

Due to the equivalence described above, absolute continuity is often discussed in the context of σ-finite measures. For example , distribution classes dominated in mathematical statistics are dealt with. A dominated class is the totality of all probability measures that are absolutely continuous with respect to a common σ-finite measure. Furthermore, the following fundamental theorems apply to σ-finite measures.

Radon-Nikodým theorem

The set of Radon Nikodým versa, the above example to the density function.

If σ-finite, then the following applies for a further measure if and only if a density has re. ${\displaystyle \mu }$ ${\displaystyle \nu \ll \mu }$ ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$

Lebesgue decomposition theorem

The decomposition theorem of Lebesgue provides the existence of a decomposition of a -endlichen measure in an absolutely continuous and a singular component. ${\displaystyle \sigma }$

If there are two σ-finite measures, then there are two more σ-finite measures and with , so that and . ${\displaystyle \mu ,\nu }$ ${\displaystyle \nu _{a}}$ ${\displaystyle \nu _{s}}$ ${\displaystyle \nu =\nu _{a}+\nu _{s}}$ ${\displaystyle \nu _{a}\ll \mu }$ ${\displaystyle \nu _{s}\perp \mu }$

literature

• Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 .
• Walter Rudin : Real and Complex Analysis . 3. Edition. McGraw-Hill, New York 1987 (English).