Radon-Nikodým theorem

In mathematics , Radon-Nikodým's theorem generalizes the derivation of a function to dimensions and signed dimensions . It provides information about when a (signed) measure can be represented by the Lebesgue integral of a function , and is of central importance for both measure and probability theory . ${\displaystyle \nu }$ ${\displaystyle f}$

The sentence is named after the Austrian mathematician Johann Radon , who proved the special case in 1913 , and the Polish Otton Marcin Nikodým , who was able to prove the general case in 1930. ${\displaystyle \mathbb {R} ^{n}}$

Preliminary remark

If a measure is on the measuring space and is a measurable function that can be integrated or quasi- integrated , then through ${\displaystyle \mu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle f\colon X\to \mathbb {R} }$ ${\displaystyle \mu }$

${\displaystyle \nu (E)=\int _{E}f\,\mathrm {d} \mu }$for all , ${\displaystyle E\in {\mathcal {A}}}$

a signed degree on defined. If it is non-negative, there is a measure. Is integrable with respect to it is finite. ${\displaystyle \nu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle f}$ ${\displaystyle \nu }$ ${\displaystyle f}$ ${\displaystyle \mu }$ ${\displaystyle \nu }$

The function is then called density function of respect . If a -zero set, that is, is , then is too . The (signed) measure is therefore absolutely constant with regard to (in characters ). ${\displaystyle f}$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle E\in {\mathcal {A}}}$ ${\displaystyle \mu }$ ${\displaystyle \mu (E)=0}$ ${\displaystyle \nu (E)=0}$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu \ll \mu }$

Radon-Nikodým's theorem states that, under certain conditions, the converse also holds:

Formulation of the sentence

Let be a σ-finite measure on the measurement space and be a σ-finite signed measure that is absolutely continuous with respect to ( ). ${\displaystyle \mu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu \ll \mu }$

Then has a density function with respect to , that is, there exists a measurable function such that ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle f\colon X\to \mathbb {R} }$

${\displaystyle \nu (E)=\int _{E}f\,\mathrm {d} \mu }$for everyone . ${\displaystyle E\in {\mathcal {A}}}$

Is another function with this property, it is true - almost everywhere with the same. If there is a measure, then it is non-negative. Is finite, then is integrable with respect to . ${\displaystyle g}$ ${\displaystyle \mu }$ ${\displaystyle f}$ ${\displaystyle \nu }$ ${\displaystyle f}$ ${\displaystyle \nu }$ ${\displaystyle f}$ ${\displaystyle \mu }$

The density function is also referred to as the Radon-Nikodým density or Radon-Nikodým derivation of regarding and is written in analogy to differential calculus as . ${\displaystyle f}$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle {\tfrac {\mathrm {d} \nu }{\mathrm {d} \mu }}}$

The theorem can be generalized to complex but generally not vectorial measures . In the case of vectorial measures, the validity depends on the Banach space used for the values ​​of the measure. The spaces for which the proposition remains valid are called spaces with the Radon-Nikodym property . ${\displaystyle \nu }$

properties

• There were , and -endliche measurements on the same measuring room. If  and ( and are absolutely continuous with respect to ) then holds ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$ ${\displaystyle \sigma }$ ${\displaystyle \nu \ll \lambda }$ ${\displaystyle \mu \ll \lambda }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$

${\displaystyle {\frac {\mathrm {d} (\nu +\mu )}{\mathrm {d} \lambda }}={\frac {\mathrm {d} \nu }{\mathrm {d} \lambda }}+{\frac {\mathrm {d} \mu }{\mathrm {d} \lambda }}}$    ${\displaystyle \lambda }$-almost everywhere.

• If is, then holds ${\displaystyle \nu \ll \mu \ll \lambda }$

${\displaystyle {\frac {\mathrm {d} \nu }{\mathrm {d} \lambda }}={\frac {\mathrm {d} \nu }{\mathrm {d} \mu }}{\frac {\mathrm {d} \mu }{\mathrm {d} \lambda }}}$    ${\displaystyle \lambda }$-almost everywhere.

• If and is an integrable function, then applies ${\displaystyle \mu \ll \lambda }$ ${\displaystyle g}$ ${\displaystyle \mu }$

${\displaystyle \int _{X}g\,\mathrm {d} \mu =\int _{X}g{\frac {\mathrm {d} \mu }{\mathrm {d} \lambda }}\,\mathrm {d} \lambda .}$

• If and is, then applies ${\displaystyle \mu \ll \nu }$ ${\displaystyle \nu \ll \mu }$

${\displaystyle {\frac {\mathrm {d} \mu }{\mathrm {d} \nu }}=\left({\frac {\mathrm {d} \nu }{\mathrm {d} \mu }}\right)^{-1}.}$

• If is a finite signed measure or a complex measure then holds ${\displaystyle \nu }$

${\displaystyle {\mathrm {d} |\nu | \over \mathrm {d} \mu }=\left|{\mathrm {d} \nu \over \mathrm {d} \mu }\right|.}$

Special case probability measures

Let it be a probability space and be too equivalent a probability measure , i. H. and . Then there exists a positive random variable such that and , where denotes the expected value with respect to. Is a real random variable then if and only if . The following applies to the expected value in this case . (For the notation, see also Lp space .) ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ ${\displaystyle Q}$ ${\displaystyle P}$ ${\displaystyle P\ll Q}$ ${\displaystyle Q\ll P}$ ${\displaystyle Z\in L^{1}(P)}$ ${\displaystyle {\tfrac {\mathrm {d} Q}{\mathrm {d} P}}=Z}$ ${\displaystyle E_{P}(Z)=1}$ ${\displaystyle E_{P}}$ ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle X\in L^{1}(Q)}$ ${\displaystyle XZ\in L^{1}(P)}$ ${\displaystyle Q}$ ${\displaystyle E_{Q}(X)=E_{P}(XZ)}$

If a probability measure is absolutely continuous on the real line with respect to the Lebesgue measure , then the Radon-Nikodým density is the probability density of , in the sense of equality, almost everywhere. In this case one calls an absolutely continuous probability distribution ; in particular then it cannot be discreet . ${\displaystyle P}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \lambda }$ ${\displaystyle {\tfrac {\mathrm {d} P}{\mathrm {d} \lambda }}}$ ${\displaystyle P}$ ${\displaystyle \lambda }$ ${\displaystyle P}$ ${\displaystyle P}$

Further statements

The decomposition theorem of Lebesgue provides a further statement in the event that not absolutely continuous with respect is. It deals with the existence and uniqueness of a decomposition of such that a part is absolutely continuous with respect to , i.e. has a density with respect to , and another part is singular with respect to . ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$

There are also formulations of the Radon-Nikodým theorem for larger classes of dimension spaces than the finite dimension spaces, the so-called decomposable dimension spaces . ${\displaystyle \sigma }$

literature

• Jürgen Elstrodt : Measure and integration theory. 7th, corrected and updated edition. Springer, Berlin et al. 2011, ISBN 978-3-642-17904-4 .
• Dirk Werner : Functional Analysis. 6th, corrected edition. Springer, Berlin et al. 2007, ISBN 978-3-540-72533-6 .