Radon–Nikodym theorem

From Freepedia

In mathematics, the Radon–Nikodym theorem is a result in functional analysis that states that if a measure ν is absolutely continuous with respect to another measure μ which is sigma-finite, then there is a measurable function f on the underlying measure space and taking values in [0,∞], such that

<math>\nu(A) = \int_A f \, d\mu</math>

for any measurable set A.

The function f is uniquely defined up to a null set, that is, if g is another function which satisfies the same property, then f = g almost everywhere. It is commonly written dν/dμ and is called the Radon–Nikodym derivative. The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of density of one measure with respect to another. A similar theorem can be proven for signed measures.

The theorem is named for Johann Radon, who proved the theorem for the special case where the underlying space is R^N in 1913, and for Otto Nikodym who proved the general case in 1930.

Properties

• Let <math>\nu</math>, <math>\mu</math>, and <math>\lambda</math> be <math>\sigma</math>-finite measures on the same measure space. If <math>\nu \ll \lambda</math> and <math>\mu\ll\lambda</math> (<math>\nu</math> and <math>\mu</math> are absolutely continuous in respect to <math>\lambda</math>), then

<math> \frac{d(\nu+\mu)}{d\lambda} = \frac{d\nu}{d\lambda}+\frac{d\mu}{d\lambda}</math> <math>\mu</math>-almost everywhere.

• If <math>\nu\ll\mu\ll\lambda</math>, then

<math> \frac{d\nu}{d\lambda}=\frac{d\nu}{d\mu}\frac{d\mu}{d\lambda}</math> <math>\mu</math>-almost everywhere.

• If <math>\mu\ll\lambda</math> and <math>g</math> is a <math>\mu</math>-integrable function, then

<math> \int_\Omega g\,d\mu = \int_\Omega g\frac{d\mu}{d\lambda}\,d\lambda.</math>

• If <math>\mu\ll\nu</math> and <math>\nu \ll\mu</math>, then

<math> \frac{d\mu}{d\nu}=\left(\frac{d\nu}{d\mu}\right)^{-1}.</math>

The assumption of sigma-finiteness

The Radon–Nikodym theorem makes the assumption that the measure μ in respect to which one computes the rate of change of ν is sigma-finite. Here is an example when μ is not sigma-finite and the Radon–Nikodym theorem fails to hold.

Consider the Borel sigma-algebra on the real line. Let the measure μ of a Borel set A be defined as the number of elements of A if A is finite, and ∞ otherwise. One can quickly check that it is a measure. It is not sigma-finite as not every Borel set is at most a countable union of finite sets. Let ν be the usual Lebesgue measure on the Borel algebra. Then, ν is absolutely continuous in respect to μ, since for a set A one has <math>\mu(A)=0</math> only if A is the empty set, and then <math>\nu(A)=0</math> is also zero.

Assume that the Radon–Nikodym theorem holds, that is, for some measurable function f one has

<math>\nu(A) = \int_A f \, d\mu</math>

for all Borel sets. Taking A to be a singleton set, A={a}, and plugging in the above equality, one finds

<math> 0 = f(a)</math>

for all real numbers a. This implies that the function f, and therefore the measure Lebesgue ν, is zero, which is a contradiction.

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This article incorporates material from Radon-Nikodym theorem on PlanetMath, which is licensed under the GFDL.