Radon–Nikodym theorem


In mathematics, the Radon–Nikodym theorem is a result in measure theory.
It involves a measurable space
on which two σ-finite measures are defined, and.
It states that, if , then there is a -measurable function, such that for any measurable set,
The function is called the Radon–Nikodym derivative and is denoted by.
The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is in 1913, and for Otto Nikodym who proved the general case in 1930. In 1936 Hans Freudenthal generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem, a result in Riesz space theory; this contains the Radon–Nikodym theorem as a special case.
If is a Banach space and the generalization of the Radon–Nikodym theorem also holds, mutatis mutandis, for functions with values in, then is said to have the Radon–Nikodym property. All Hilbert spaces have the Radon–Nikodym property.

Radon–Nikodym derivative

The function satisfying the above equality is uniquely defined up to a -null set, that is, if is another function which satisfies the same property, then -almost everywhere. is commonly written 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 and complex measures: namely, that if is a nonnegative σ-finite measure, and is a finite-valued signed or complex measure such that, i.e. is absolutely continuous with respect to, then there is a -integrable real- or complex-valued function on such that for every measurable set,

Examples

In the following examples, the set is the real interval , and is the Borel sigma-algebra on.
  1. is the length measure on. assigns to each subset of, twice the length of. Then,.
  2. is the length measure on. assigns to each subset of, the number of points from the set that are contained in. Then, is not absolutely-continuous with respect to since it assigns non-zero measure to zero-length points. Indeed, there is no derivative : there is no finite function that, when integrated e.g. from to, gives for all.
  3. , where is the length measure on X and is the Dirac measure on 0. Then, is absolutely continuous with respect to, and – the derivative is 0 at and 1 at.

    Applications

The theorem is very important in extending the ideas of probability theory from probability masses and probability densities defined over real numbers to probability measures defined over arbitrary sets. It tells if and how it is possible to change from one probability measure to another. Specifically, the probability density function of a random variable is the Radon–Nikodym derivative of the induced measure with respect to some base measure.
For example, it can be used to prove the existence of conditional expectation for probability measures. The latter itself is a key concept in probability theory, as conditional probability is just a special case of it.
Amongst other fields, financial mathematics uses the theorem extensively, in particular via the Girsanov theorem. Such changes of probability measure are the cornerstone of the rational pricing of derivatives and are used for converting actual probabilities into those of the risk neutral probabilities.

Properties

Information divergences

If μ and ν are measures over, and μ ≪ ν
The Radon–Nikodym theorem makes the assumption that the measure μ with respect to which one computes the rate of change of ν is σ-finite. Here is an example when μ is not σ-finite and the Radon–Nikodym theorem fails to hold.
Consider the Borel σ-algebra on the real line. Let the counting measure,, of a Borel set be defined as the number of elements of if is finite, and otherwise. One can check that is indeed a measure. It is not -finite, as not every Borel set is at most a countable union of finite sets. Let be the usual Lebesgue measure on this Borel algebra. Then, is absolutely continuous with respect to, since for a set one has only if is the empty set, and then is also zero.
Assume that the Radon–Nikodym theorem holds, that is, for some measurable function one has
for all Borel sets. Taking to be a singleton set, and using the above equality, one finds
for all real numbers. This implies that the function, and therefore the Lebesgue measure, is zero, which is a contradiction.

Proof

This section gives a measure-theoretic proof of the theorem. There is also a functional-analytic proof, using Hilbert space methods, that was first given by von Neumann.
For finite measures and, the idea is to consider functions with. The supremum of all such functions, along with the monotone convergence theorem, then furnishes the Radon–Nikodym derivative. The fact that the remaining part of is singular with respect to follows from a technical fact about finite measures. Once the result is established for finite measures, extending to -finite, signed, and complex measures can be done naturally. The details are given below.

For finite measures

First, suppose and are both finite-valued nonnegative measures. Let be the set of those measurable functions such that:
, since it contains at least the zero function. Now let, and suppose is an arbitrary measurable set, and define:
Then one has
and therefore,.
Now, let be a sequence of functions in such that
By replacing with the maximum of the first functions, one can assume that the sequence is increasing. Let be an extended-valued function defined as
By Lebesgue's monotone convergence theorem, one has
for each, and hence,. Also, by the construction of,
Now, since,
defines a nonnegative measure on. Suppose ; then, since is finite, there is an such that. Let be a Hahn decomposition for the signed measure. Note that for every one has, and hence,
where is the indicator function of. Also, note that ; for if, then , so and
contradicting the fact that.
Then, since
and satisfies
This is impossible; therefore, the initial assumption that must be false. Hence,, as desired.
Now, since is -integrable, the set is -null. Therefore, if a is defined as
then has the desired properties.
As for the uniqueness, let be measurable functions satisfying
for every measurable set. Then, is -integrable, and
In particular, for or It follows that
and so, that -almost everywhere; the same is true for, and thus, -almost everywhere, as desired.

For -finite positive measures

If and are -finite, then can be written as the union of a sequence of disjoint sets in, each of which has finite measure under both and. For each, by the finite case, there is a -measurable function such that
for each -measurable subset of. The sum of those functions is then the required function such that.
As for the uniqueness, since each of the is -almost everywhere unique, then so is.

For signed and complex measures

If is a -finite signed measure, then it can be Hahn–Jordan decomposed as where one of the measures is finite. Applying the previous result to those two measures, one obtains two functions,, satisfying the Radon–Nikodym theorem for and respectively, at least one of which is -integrable. It is clear then that satisfies the required properties, including uniqueness, since both and are unique up to -almost everywhere equality.
If is a complex measure, it can be decomposed as, where both and are finite-valued signed measures. Applying the above argument, one obtains two functions,, satisfying the required properties for and, respectively. Clearly, is the required function.

The Lebesgue decomposition theorem

shows that the assumptions of the Radon-Nikodym theorem can be found even in a situation which is seemingly more general. Consider a σ-finite positive measure on the measure space and a σ-finite signed measure on, without assuming any absolute continuity. Then there exist unique signed measures and on such that,, and. The Radon-Nikodym theorem can then be applied to the pair.