Radon–Nikodym theorem

In mathematics, the Radon - Nikodým generalizes the derivative of a function on dimensions and signed extent. He then tells you when a ( signed ) measure is represented by the Lebesgue integral of a function, and is both the measure and the probability theory of central importance.

Named the sentence after the Austrian mathematician Johann Radon, who in 1913 proved a special case, and after Otton Marcin Nikodým, who proved the general case 1930.

Preliminary

Is a measure on the measuring space and is an integrable or quasi- integrable with respect to measurable function, then by

A signed measure on defined. Is non- negative, is a measure. Is integrable with respect, so is finite.

The function is then called density function of respect. Is a -null set, that is, so is. The (signed ) measure is therefore absolutely continuous with respect to (in characters).

The set of Radon Nikodým states that the converse also holds under certain conditions:

Wording of the sentence

Be a σ - finite measure on the measuring space and be a signed measure which is absolutely continuous with respect to ().

Then has a density function with respect to, that is, there exists a measurable function, so that

If another function with this property, so it is true -almost everywhere agreement with. Is a measure, it is not negative. Is finite, so is integrable with respect.

The density function is also known as radon Nikodým density or Radon Nikodým derivation of respect and written in analogy to the differential calculus as.

Properties

• Let ν, μ, and λ σ - finite measures on the same measurement space. If ν « λ and μ « λ ( ν and μ are absolutely continuous with respect to λ ), then

• If ν « μ « is λ, then applies

• If μ « λ and g is a μ - integrable function, then applies

• If μ « ν and ν « μ, then applies

• If ν is a finite signed measure or a complex measure, then applies

Special case of probability measures

There is a probability space and let a be equivalent probability measure, that is, and. Then there exists a positive random variable such that and, where denotes the expectation value with respect. Is a real random variable, it is precisely when. The expectation value with respect to true in this case.