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 .
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.
Preliminary remark
If a measure is on the measuring space and is a measurable function that can be integrated or quasi- integrated , then through
- for all ,
a signed degree on defined. If it is non-negative, there is a measure. Is integrable with respect to it is finite.
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 ).
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 ( ).
Then has a density function with respect to , that is, there exists a measurable function such that
- for everyone .
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 .
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 .
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 .
properties
- There were , and -endliche measurements on the same measuring room. If and ( and are absolutely continuous with respect to ) then holds
- -almost everywhere.
- If is, then holds
- -almost everywhere.
- If and is an integrable function, then applies
- If and is, then applies
- If is a finite signed measure or a complex measure then holds
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 .)
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 .
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 .
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 .
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 .