Radon–Nikodym theorem

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In mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure Template:Mvar on , then there is a measurable function , such that for any measurable subset :

The function f is called the Radon–Nikodym derivative and denoted by .

The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is RN in 1913, and for Otto Nikodym who proved the general case in 1930.[1] In 1936 Hans Freudenthal further generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem, a result in Riesz space theory, which contains the Radon–Nikodym theorem as a special case.[2]

If Template:Mvar is a Banach space and the generalization of the Radon–Nikodym theorem also holds for functions with values in Template:Mvar (mutatis mutandis), then Template:Mvar is said to have the Radon–Nikodym property. All Hilbert spaces have the Radon–Nikodym property.

Radon–Nikodym derivative

The function f satisfying the above equality is uniquely defined up to a Template:Mvar-null set, that is, if Template:Mvar is another function which satisfies the same property, then f = g Template:Mvar-almost everywhere. f 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 (the way the Jacobian determinant is used in multivariable integration). A similar theorem can be proven for signed and complex measures: namely, that if Template:Mvar is a nonnegative σ-finite measure, and ν is a finite-valued signed or complex measure such that ν ≪ μ, i.e. ν is absolutely continuous with respect to Template:Mvar, then there is a Template:Mvar-integrable real- or complex-valued function Template:Mvar on Template:Mvar such that for every measurable set Template:Mvar,

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 (usually the Lebesgue measure for continuous random variables).

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. 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

  • Let ν, μ, and λ be σ-finite measures on the same measure space. If ν ≪ λ and μ ≪ λ (ν and μ are absolutely continuous in respect to λ, then
  • If ν ≪ μ ≪ λ, then
  • In particular, if μ ≪ ν and ν ≪ μ, then
  • If ν is a finite signed or complex measure, then

Further applications

Information divergences

If μ and ν are measures over Template:Mvar, and μ ≪ ν

  • For α > 0, α ≠ 1 the Rényi divergence of order α from μ to ν is defined to be

The assumption of σ-finiteness

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, Template:Mvar, of a Borel set Template:Mvar be defined as the number of elements of Template:Mvar if Template:Mvar is finite, and otherwise. One can check that Template:Mvar is indeed a measure. It is not Template:Mvar-finite, as not every Borel set is at most a countable union of finite sets. Let Template:Mvar be the usual Lebesgue measure on this Borel algebra. Then, Template:Mvar is absolutely continuous with respect to Template:Mvar, since for a set Template:Mvar one has μ(A) = 0 only if Template:Mvar is the empty set, and then ν(A) is also zero.

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

for all Borel sets. Taking Template:Mvar to be a singleton set, A = {a}, and using the above equality, one finds

for all real numbers Template:Mvar. This implies that the function f, and therefore the Lebesgue measure Template:Mvar, 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 Template:Mvar and Template:Mvar, the idea is to consider functions f with f dμ. 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 Template:Mvar is singular with respect to Template:Mvar follows from a technical fact about finite measures. Once the result is established for finite measures, extending to Template:Mvar-finite, signed, and complex measures can be done naturally. The details are given below.

For finite measures

First, suppose Template:Mvar and Template:Mvar are both finite-valued nonnegative measures. Let Template:Mvar be the set of those measurable functions f  : X → [0, ∞) such that:

F ≠ ∅, since it contains at least the zero function. Now let f1,  f2F, and suppose Template:Mvar be an arbitrary measurable set, and define:

Then one has

and therefore, max{ f1,  f2} ∈ F.

Now, let { fn } be a sequence of functions in Template:Mvar such that

By replacing fn with the maximum of the first Template:Mvar functions, one can assume that the sequence { fn } is increasing. Let Template:Mvar be a function defined as

By Lebesgue's monotone convergence theorem, one has

for each A ∈ Σ, and hence, gF. Also, by the construction of Template:Mvar,

Now, since gF,

defines a nonnegative measure on Σ. Suppose ν0 ≠ 0; then, since Template:Mvar is finite, there is an ε > 0 such that ν0(X) > ε μ(X). Let (PN) be a Hahn decomposition for the signed measure ν0ε μ. Note that for every A ∈ Σ one has ν0(AP) ≥ ε μ(AP), and hence,

Also, note that μ(P) > 0; for if μ(P) = 0, then (since Template:Mvar is absolutely continuous in relation to Template:Mvar) ν0(P) ≤ ν(P) = 0, so ν0(P) = 0 and

contradicting the fact that ν0(X) > εμ(X).

Then, since

g + ε 1PF and satisfies

This is impossible, therefore, the initial assumption that ν0 ≠ 0 must be false. So ν0 = 0, as desired.

Now, since Template:Mvar is Template:Mvar-integrable, the set {xX : g(x) = ∞} is Template:Mvar-null. Therefore, if a f is defined as

then f has the desired properties.

As for the uniqueness, let f, g : X → [0, ∞) be measurable functions satisfying

for every measurable set Template:Mvar. Then, gf is Template:Mvar-integrable, and

In particular, for A = {xX : f(x) > g(x)}, or {xX : f(x) < g(x)}. It follows that

and so, that (gf )+ = 0 Template:Mvar-almost everywhere; the same is true for (gf ), and thus, f  = g Template:Mvar-almost everywhere, as desired.

For Template:Mvar-finite positive measures

If Template:Mvar and Template:Mvar are Template:Mvar-finite, then Template:Mvar can be written as the union of a sequence {Bn}n of disjoint sets in Σ, each of which has finite measure under both Template:Mvar and Template:Mvar. For each Template:Mvar, there is a Σ-measurable function fn  : Bn → [0, ∞) such that

for each Σ-measurable subset Template:Mvar of Bn. The union f of those functions is then the required function.

As for the uniqueness, since each of the fn is Template:Mvar-almost everywhere unique, then so is f.

For signed and complex measures

If Template:Mvar is a Template:Mvar-finite signed measure, then it can be Hahn–Jordan decomposed as ν = ν+ν where one of the measures is finite. Applying the previous result to those two measures, one obtains two functions, g, h : X → [0, ∞), satisfying the Radon–Nikodym theorem for ν+ and ν respectively, at least one of which is Template:Mvar-integrable (i.e., its integral with respect to Template:Mvar is finite). It is clear then that f = gh satisfies the required properties, including uniqueness, since both Template:Mvar and Template:Mvar are unique up to Template:Mvar-almost everywhere equality.

If Template:Mvar is a complex measure, it can be decomposed as ν = ν1 + 2, where both ν1 and ν2 are finite-valued signed measures. Applying the above argument, one obtains two functions, g, h : X → [0, ∞), satisfying the required properties for ν1 and ν2, respectively. Clearly, f  = g + ih is the required function.

See also

Notes

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References

  • Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8.
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  • Royden H.L 1965 real Analysis , Macmillan - contains a lucid proof in case the measure ν is not σ-finite.
  • Lang Serge 1965 " analysis II -real analysius " contains a proof for vector measures assuming values in a Banach space.

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