Editing Radon–Nikodym theorem

{{Short description|Expressing a measure as an integral of another}}
In [[mathematics]], the '''Radon–Nikodym theorem''' is a result in [[measure theory]] that expresses the relationship between two measures defined on the same [[measurable space]]. A ''measure'' is a [[set function]] that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider [[probability space]].

One way to derive a new measure from one already given is to assign a density to each point of the space, then [[Lebesgue integration|integrate]] over the measurable subset of interest. This can be expressed as

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

where {{math|''ν''}} is the new measure being defined for any measurable subset {{math|''A''}} and the function {{math|''f''}} is the density at a given point. The integral is with respect to an existing measure {{math|''μ''}}, which may often be the canonical [[Lebesgue measure]] on the [[real line]] {{math|'''R'''}} or the ''n''-dimensional [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} (corresponding to our standard notions of length, area and volume). For example, if {{math|''f''}} represented mass density and {{math|''μ''}} was the Lebesgue measure in three-dimensional space {{math|'''R'''<sup>3</sup>}}, then {{math|''ν''(''A'')}} would equal the total mass in a spatial region {{math|''A''}}.

The Radon–Nikodym theorem essentially states that, under certain conditions, any measure {{math|''ν''}} can be expressed in this way with respect to another measure {{math|''μ''}} on the same space. The function {{math| ''f'' }} is then called the '''Radon–Nikodym derivative''' and is denoted by <math>\tfrac{d\nu}{d\mu}</math>.<ref>{{cite book |first=Patrick |last=Billingsley |title=Probability and Measure |location=New York |publisher=John Wiley & Sons |edition=Third |year=1995 |isbn=0-471-00710-2 |pages=419–427 }}</ref> An important application is in [[probability theory]], leading to the [[probability density function]] of a [[random variable]].

The theorem is named after [[Johann Radon]], who proved the theorem for the special case where the underlying space is {{math|'''R'''<sup>''n''</sup>}} in 1913, and for [[Otto Nikodym]] who proved the general case in 1930.<ref>{{cite journal
 |last=Nikodym |first=O. |author-link=Otton Nikodym |language=fr |title=Sur une généralisation des intégrales de M. J. Radon
 |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm15/fm15114.pdf |access-date=2018-01-30 |journal=[[Fundamenta Mathematicae]] |year=1930 |volume=15 |pages=131–179 |jfm=56.0922.02|doi=10.4064/fm-15-1-131-179 |doi-access=free }}</ref> 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.<ref>{{Cite book| last = Zaanen  | first = Adriaan C. | author-link= Adriaan Cornelis Zaanen| year = 1996  | title = Introduction to Operator Theory in Riesz Spaces | publisher = [[Springer Science+Business Media|Springer]] | isbn = 3-540-61989-5 }}</ref>

A [[Banach space]] {{mvar|Y}} is said to have the [[Radon–Nikodym property]] if the generalization of the Radon–Nikodym theorem also holds, ''[[mutatis mutandis]]'', for functions with values in {{mvar|Y}}. All [[Hilbert space]]s have the Radon–Nikodym property.

==Formal description==
===Radon–Nikodym theorem===
The '''Radon–Nikodym theorem''' involves a [[measurable space]] <math>(X, \Sigma)</math> on which two [[σ-finite measure]]s are defined, <math>\mu</math> and <math>\nu.</math> 
It states that, if <math>\nu \ll \mu</math> (that is, if <math>\nu</math> is [[Absolute continuity#Absolute continuity of measures|absolutely continuous]] with respect to <math>\mu</math>), then there exists a <math>\Sigma</math>-[[measurable function]] <math>f : X \to [0, \infty),</math> such that for any measurable set <math>A \in \Sigma,</math>
<math display=block>\nu(A) = \int_A f \, d\mu.</math>

===Radon–Nikodym derivative===
The function <math>f</math> satisfying the above equality is {{em|uniquely defined [[up to]] a <math>\mu</math>-[[null set]]}}, that is, if <math>g</math> is another function which satisfies the same property, then <math>f = g</math> {{nowrap|<math>\mu</math>-[[almost everywhere]]}}. The function <math>f</math> is commonly written <math display>\frac{d\nu}{d\mu}</math> and is called the '''{{visible anchor|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).

===Extension to signed or complex measures===
A similar theorem can be proven for [[Signed measure|signed]] and [[complex measure]]s: namely, that if <math>\mu</math> is a nonnegative σ-finite measure, and <math>\nu</math> is a finite-valued signed or complex measure such that <math>\nu \ll \mu,</math> that is, <math>\nu</math> is [[absolutely continuous]] with respect to <math>\mu,</math> then there is a <math>\mu</math>-integrable real- or complex-valued function <math>g</math> on <math>X</math> such that for every measurable set <math>A,</math>
<math display=block>\nu(A) = \int_A g \, d\mu.</math>

== Examples ==
In the following examples, the set {{mvar|X}} is the real interval [0,1], and <math>\Sigma</math> is the [[Borel set|Borel sigma-algebra]] on {{mvar|X}}.
# <math>\mu</math> is the length measure on {{mvar|X}}. <math>\nu</math> assigns to each subset {{mvar|Y}} of {{mvar|X}}, twice the length of {{mvar|Y}}. Then, <math display="inline">\frac{d\nu}{d\mu} = 2</math>.
# <math>\mu</math> is the length measure on {{mvar|X}}. <math>\nu</math> assigns to each subset {{mvar|Y}} of {{mvar|X}}, the number of points from the set {0.1, …, 0.9} that are contained in {{mvar|Y}}. Then, <math>\nu</math> is not absolutely-continuous with respect to <math>\mu</math> since it assigns non-zero measure to zero-length points. Indeed, there is no derivative <math display="inline">\frac{d\nu}{d\mu}</math>: there is no finite function that, when integrated e.g. from <math>(0.1 - \varepsilon)</math> to <math>(0.1 + \varepsilon)</math>, gives <math>1</math> for all <math>\varepsilon > 0</math>.
# <math>\mu = \nu + \delta_0</math>, where <math>\nu</math> is the length measure on {{mvar|X}} and <math>\delta_0</math> is the [[Dirac measure]] on 0 (it assigns a measure of 1 to any set containing 0 and a measure of 0 to any other set). Then, <math>\nu</math> is absolutely continuous with respect to <math>\mu</math>, and <math display="inline">\frac{d\nu}{d\mu} = 1_{X\setminus \{0\}}</math> – the derivative is 0 at <math>x = 0</math> and 1 at <math>x > 0</math>.<ref>{{cite web |title=Calculating Radon Nikodym derivative |date=April 7, 2018 |work=[[Stack Exchange]] |url=https://math.stackexchange.com/q/588442/29780 }}</ref>

==Properties==
* Let ''ν'', ''μ'', and ''λ'' be σ-finite measures on the same measurable space. If ''ν'' ≪ ''λ'' and ''μ'' ≪ ''λ'' (''ν'' and ''μ'' are both [[Absolutely continuous#Absolute continuity of measures|absolutely continuous]] with respect to ''λ''), then <math display="block"> \frac{d(\nu+\mu)}{d\lambda} = \frac{d\nu}{d\lambda}+\frac{d\mu}{d\lambda} \quad \lambda\text{-almost everywhere}.</math>
* If ''ν'' ≪ ''μ'' ≪ ''λ'', then <math display="block">\frac{d\nu}{d\lambda}=\frac{d\nu}{d\mu}\frac{d\mu}{d\lambda}\quad\lambda\text{-almost everywhere}.</math>
* In particular, if ''μ'' ≪ ''ν'' and ''ν'' ≪ ''μ'', then <math display="block"> \frac{d\mu}{d\nu}=\left(\frac{d\nu}{d\mu}\right)^{-1}\quad\nu\text{-almost everywhere}.</math>
* If ''μ'' ≪ ''λ'' and {{mvar|g}} is a ''μ''-integrable function, then <math display="block"> \int_X g\,d\mu = \int_X g\frac{d\mu}{d\lambda}\,d\lambda.</math>
* If ''ν'' is a finite signed or complex measure, then <math display="block"> {d|\nu|\over d\mu} = \left|{d\nu\over d\mu}\right|. </math>

==Applications==
===Probability theory===
The theorem is very important in extending the ideas of [[probability theory]] from probability masses and probability densities defined over real numbers to [[probability measure]]s 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 variable]]s).

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.

===Financial mathematics===
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 [[Derivative (finance)|derivatives]] and are used for converting actual probabilities into those of the [[risk-neutral measure|risk neutral probabilities]].

===Information divergences===
If ''μ'' and ''ν'' are measures over {{mvar|X}}, and ''μ'' ≪ ''ν''

* The [[Kullback–Leibler divergence]] from ''ν'' to ''μ'' is defined to be <math display="block"> D_\text{KL}(\mu \parallel \nu) = \int_X \log \left( \frac{d \mu}{d \nu} \right) \; d\mu.</math>
* For ''α'' > 0, ''α'' ≠ 1 the [[Renyi divergence|Rényi divergence]] of order ''α'' from ''ν'' to ''μ'' is defined to be <math display="block">D_\alpha(\mu \parallel \nu) = \frac{1}{\alpha - 1} \log\left(\int_X\left(\frac{d\mu}{d\nu}\right)^{\alpha-1}\; d\mu\right).</math>

==The assumption of σ-finiteness==
The Radon–Nikodym theorem above makes the assumption that the measure ''μ'' with respect to which one computes the rate of change of ''ν'' is [[σ-finite measure|σ-finite]].

===Negative example===
Here is an example when ''μ'' is not σ-finite and the Radon–Nikodym theorem fails to hold.

Consider the [[Borel algebra|Borel σ-algebra]] on the [[real line]]. Let the [[counting measure]], {{mvar|μ}}, of a Borel set {{mvar|A}} be defined as the number of elements of {{mvar|A}} if {{mvar|A}} is finite, and {{math|∞}} otherwise. One can check that {{mvar|μ}} is indeed a measure. It is not {{mvar|σ}}-finite, as not every Borel set is at most a countable union of finite sets. Let {{mvar|ν}} be the usual [[Lebesgue measure]] on this Borel algebra. Then, {{mvar|ν}} is absolutely continuous with respect to {{mvar|μ}}, since for a set {{mvar|A}} one has {{math|1=''μ''(''A'') = 0}} only if {{mvar|A}} is the [[empty set]], and then {{math|''ν''(''A'')}} is also zero.

Assume that the Radon–Nikodym theorem holds, that is, for some measurable function {{math|''f''}} one has
:<math>\nu(A) = \int_A f \,d\mu</math>

for all Borel sets. Taking {{mvar|A}} to be a [[singleton set]], {{math|1=''A'' = {''a''}<nowiki/>}}, and using the above equality, one finds
:<math> 0 = f(a)</math>

for all real numbers {{mvar|a}}. This implies that the function {{math| ''f'' }}, and therefore the Lebesgue measure {{mvar|ν}}, is zero, which is a contradiction.

===Positive result===
Assuming <math>\nu\ll\mu,</math> the Radon–Nikodym theorem also holds if <math>\mu</math> is [[Measure (mathematics)#Localizable measures|localizable]] and <math>\nu</math> is ''accessible with respect to'' <math>\mu</math>,<ref name=BP>{{cite book |last1=Brown |first1=Arlen |last2=Pearcy |first2=Carl |title=Introduction to Operator Theory I: Elements of Functional Analysis |date=1977 |isbn=978-1461299288}}</ref>{{rp|at=p. 189, Exercise 9O}} i.e., <math>\nu(A)=\sup\{\nu(B):B\in{\cal P}(A)\cap\mu^\operatorname{pre}(\R_{\ge0})\}</math> for all <math>A\in\Sigma.</math><ref>{{cite book |last1=Fonseca |first1=Irene |last2=Leoni |first2=Giovanni |title=Modern Methods in the Calculus of Variations: L<sup>p</sup> Spaces |publisher=Springer |isbn=978-0-387-35784-3 |page=68}}</ref>{{rp|at=Theorem 1.111 (Radon–Nikodym, II)}}<ref name=BP/>{{rp|at=p. 190, Exercise 9T(ii)}}

==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 [[John von Neumann|von Neumann]].

For finite measures {{mvar|μ}} and {{mvar|ν}}, the idea is to consider functions {{math| ''f'' }} with {{math|''f dμ'' ≤ ''dν''}}. 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 {{mvar|μ}} is singular with respect to {{mvar|ν}} follows from a technical fact about finite measures. Once the result is established for finite measures, extending to {{mvar|σ}}-finite, signed, and complex measures can be done naturally. The details are given below.

===For finite measures===

'''Constructing an extended-valued candidate'''  First, suppose {{mvar|μ}} and {{mvar|ν}} are both finite-valued nonnegative measures. Let {{mvar|F}} be the set of those extended-value measurable functions {{math|''f''  : ''X'' → [0, ∞]}} such that:
:<math>\forall A \in \Sigma:\qquad \int_A f\,d\mu \leq \nu(A)</math>

{{math|''F'' ≠ ∅}}, since it contains at least the zero function. Now let {{math|''f''<sub>1</sub>,  ''f''<sub>2</sub> ∈ ''F''}}, and suppose {{mvar|A}} is an arbitrary measurable set, and define:
:<math>\begin{align}
  A_1 &= \left\{ x \in A : f_1(x) > f_2(x) \right\}, \\
  A_2 &= \left\{ x \in A : f_2(x) \geq f_1(x) \right\}.
\end{align}</math>

Then one has
:<math>\int_A\max\left\{f_1, f_2\right\}\,d\mu = \int_{A_1} f_1\,d\mu + \int_{A_2} f_2\,d\mu \leq \nu\left(A_1\right) + \nu\left(A_2\right) = \nu(A),</math>

and therefore, {{math|max{ ''f'' <sub>1</sub>,  ''f'' <sub>2</sub>} ∈ ''F''}}.

Now, let {{math|{{mset| ''f<sub>n</sub>'' }}}} be a sequence of functions in {{mvar|F}} such that
:<math>\lim_{n\to\infty}\int_X f_n\,d\mu = \sup_{f\in F} \int_X f\,d\mu.</math>

By replacing {{math| ''f<sub>n</sub>'' }} with the maximum of the first {{mvar|n}} functions, one can assume that the sequence {{math|{{mset| ''f<sub>n</sub>'' }}}} is increasing. Let {{mvar|g}} be an extended-valued function defined as
:<math>g(x) := \lim_{n\to\infty}f_n(x).</math>

By Lebesgue's [[monotone convergence theorem]], one has
:<math>\lim_{n\to\infty} \int_A f_n\,d\mu = \int_A \lim_{n\to\infty} f_n(x)\,d\mu(x) = \int_A g\,d\mu \leq \nu(A)</math>

for each {{math|''A'' ∈ Σ}}, and hence, {{math|''g'' ∈ ''F''}}. Also, by the construction of {{mvar|g}},
:<math>\int_X g\,d\mu = \sup_{f\in F}\int_X f\,d\mu.</math>

'''Proving equality'''  Now, since {{math|''g'' ∈ ''F''}},
:<math>\nu_0(A) := \nu(A) - \int_A g\,d\mu</math>

defines a nonnegative measure on {{math|Σ}}. To prove equality, we show that {{math|1=''ν''<sub>0</sub> = 0}}.

Suppose {{math|ν<sub>0</sub> ≠ 0}}; then, since {{mvar|μ}} is finite, there is an {{math|''ε'' > 0}} such that {{math|''ν''<sub>0</sub>(''X'') > ''ε μ''(''X'')}}. To derive a contradiction from {{math|ν<sub>0</sub> ≠ 0}}, we look for a [[Positive and negative sets|positive set]] {{math|''P'' ∈ Σ}} for the signed measure {{math|''ν''<sub>0</sub> − ''ε μ''}} (i.e. a measurable set {{mvar|P}}, all of whose measurable subsets have non-negative {{math|''ν''<sub>0</sub>&nbsp;−&nbsp;''εμ''}} measure), where also {{mvar|P}} has positive {{mvar|μ}}-measure. Conceptually, we're looking for a set {{mvar|P}}, where {{math|''ν''<sub>0</sub> ≥ ''ε μ''}} in every part of {{mvar|P}}. A convenient approach is to use the [[Hahn decomposition theorem|Hahn decomposition]] {{math|(''P'', ''N'')}} for the signed measure {{math|''ν''<sub>0</sub> − ''ε μ''}}.

Note then that for every {{math|''A'' ∈ Σ}} one has {{math|''ν''<sub>0</sub>(''A'' ∩ ''P'') ≥ ''ε μ''(''A'' ∩ ''P'')}}, and hence,

:<math>\begin{align}
  \nu(A) &=    \int_A g\,d\mu + \nu_0(A) \\
         &\geq \int_A g\,d\mu + \nu_0(A\cap P)\\
         &\geq \int_A g\,d\mu + \varepsilon\mu(A\cap P)
          =    \int_A\left(g + \varepsilon 1_P\right)\,d\mu,
\end{align}</math>

where {{math|1<sub>''P''</sub>}} is the [[indicator function]] of {{mvar|P}}. Also, note that {{math|''μ''(''P'') > 0}} as desired; for if {{math|1=''μ''(''P'') = 0}}, then (since {{mvar|ν}} is absolutely continuous in relation to {{mvar|μ}}) {{math|1=''ν''<sub>0</sub>(''P'') ≤ ''ν''(''P'') = 0}}, so {{math|1=''ν''<sub>0</sub>(''P'') = 0}} and
:<math>\nu_0(X) - \varepsilon\mu(X) = \left(\nu_0 - \varepsilon\mu\right)(N) \leq 0,</math>
contradicting the fact that {{math|''ν''<sub>0</sub>(''X'') > ''εμ''(''X'')}}.

Then, since also
:<math>\int_X\left(g + \varepsilon1_P\right)\,d\mu \leq \nu(X) < +\infty,</math>

{{math|''g'' + ''ε'' 1<sub>''P''</sub> ∈ ''F''}} and satisfies
:<math>\int_X\left(g + \varepsilon 1_P\right)\,d\mu > \int_X g\,d\mu = \sup_{f\in F}\int_X f\,d\mu.</math>

This is [[reductio ad absurdum|impossible]] because it violates the definition of a [[supremum]]; therefore, the initial assumption that {{math|''ν''<sub>0</sub> ≠ 0}} must be false. Hence, {{math|1=''ν''<sub>0</sub> = 0}}, as desired.

'''Restricting to finite values'''  Now, since {{mvar|g}} is {{mvar|μ}}-integrable, the set {{math|{{mset|1=''x'' ∈ ''X'' : ''g''(''x'') = ∞}}}} is {{mvar|μ}}-[[null set|null]]. Therefore, if a {{math| ''f'' }} is defined as
:<math>f(x) = \begin{cases}
  g(x) & \text{if }g(x) < \infty \\
  0    & \text{otherwise,}
\end{cases}</math>

then {{math|''f''}} has the desired properties.

'''Uniqueness'''  As for the uniqueness, let {{math| ''f'', ''g'' : ''X'' → [0, ∞)}} be measurable functions satisfying
:<math>\nu(A) = \int_A f\,d\mu = \int_A g\,d\mu</math>

for every measurable set {{mvar|A}}. Then, {{math|''g'' − ''f'' }} is {{mvar|μ}}-integrable, and
:<math>\int_A(g - f)\,d\mu = 0.</math> (Recall that we can split the integral into two as long as they are measurable and non-negative)

In particular, for {{math|1=''A'' = {''x'' ∈ ''X'' : ''f''(''x'') > ''g''(''x'')},}} or {{math|{{mset|''x'' ∈ ''X'' : ''f''(''x'') < ''g''(''x'')}}}}. It follows that
:<math>\int_X(g - f)^+\,d\mu = 0 = \int_X(g - f)^-\,d\mu,</math>

and so, that {{math|1=(''g'' − ''f'' )<sup>+</sup> = 0}} {{mvar|μ}}-almost everywhere; the same is true for {{math|(''g'' − ''f'' )<sup>−</sup>}}, and thus, {{math|1=''f'' = ''g''}} {{mvar|μ}}-almost everywhere, as desired.

===For {{mvar|σ}}-finite positive measures===
If {{mvar|μ}} and {{mvar|ν}} are {{mvar|σ}}-finite, then {{mvar|X}} can be written as the union of a sequence {{math|{''B<sub>n</sub>''}<sub>''n''</sub>}} of [[disjoint sets]] in {{math|Σ}}, each of which has finite measure under both {{mvar|μ}} and {{mvar|ν}}. For each {{mvar|n}}, by the finite case, there is a {{math|Σ}}-measurable function {{math| ''f<sub>n</sub>''  : ''B<sub>n</sub>'' → [0, ∞)}} such that
:<math>\nu_n(A) = \int_A f_n\,d\mu</math>

for each {{math|Σ}}-measurable subset {{mvar|A}} of {{math|''B<sub>n</sub>''}}. The sum <math display="inline">\left(\sum_n f_n 1_{B_n}\right) := f</math> of those functions is then the required function such that <math display="inline">\nu(A) = \int_A f \, d\mu</math>. 

As for the uniqueness, since each of the {{math|''f<sub>n</sub>''}} is {{mvar|μ}}-almost everywhere unique, so is {{math|''f''}}.

===For signed and complex measures===
If {{mvar|ν}} is a {{mvar|σ}}-finite signed measure, then it can be Hahn–Jordan decomposed as {{math|1=''ν'' = ''ν''<sup>+</sup> − ''ν''<sup>−</sup>}} where one of the measures is finite. Applying the previous result to those two measures, one obtains two functions, {{math|''g'', ''h'' : ''X'' → [0, ∞)}}, satisfying the Radon–Nikodym theorem for {{math|''ν''<sup>+</sup>}} and {{math|''ν''<sup>−</sup>}} respectively, at least one of which is {{mvar|μ}}-integrable (i.e., its integral with respect to {{mvar|μ}} is finite). It is clear then that {{math|1=''f'' = ''g'' − ''h''}} satisfies the required properties, including uniqueness, since both {{mvar|g}} and {{mvar|h}} are unique up to {{mvar|μ}}-almost everywhere equality.

If {{mvar|ν}} is a [[complex measure]], it can be decomposed as {{math|1=''ν'' = ''ν''<sub>1</sub> + ''iν''<sub>2</sub>}}, where both {{math|''ν''<sub>1</sub>}} and {{math|''ν''<sub>2</sub>}} are finite-valued signed measures. Applying the above argument, one obtains two functions, {{math|''g'', ''h'' : ''X'' → [0, ∞)}}, satisfying the required properties for {{math|''ν''<sub>1</sub>}} and {{math|''ν''<sub>2</sub>}}, respectively. Clearly, {{math|1=''f'' = ''g'' + ''ih''}} is the required function.

==The Lebesgue decomposition theorem==
[[Lebesgue's 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 <math>\mu</math> on the measure space <math>(X,\Sigma)</math> and a σ-finite signed measure <math>\nu</math> on <math>\Sigma</math>, without assuming any absolute continuity. Then there exist unique signed measures <math>\nu_a</math> and <math>\nu_s</math> on <math>\Sigma</math> such that <math>\nu=\nu_a+\nu_s</math>, <math>\nu_a\ll\mu</math>, and <math>\nu_s\perp\mu</math>. The Radon–Nikodym theorem can then be applied to the pair <math>\nu_a,\mu</math>.

==See also==
*[[Girsanov theorem]]
*[[Radon–Nikodym set]]

==Notes==
{{Reflist}}

==References==
* {{cite book |last=Lang |first=Serge |year=1969 |title=Analysis II: Real analysis |author-link= Serge Lang |publisher=Addison-Wesley }} Contains a proof for vector measures assuming values in a Banach space.
* {{cite book |last1=Royden |first1=H. L. | last2= Fitzpatrick | first2 = P. M.| author1-link= Halsey Royden |year=2010 |title=Real Analysis |edition= 4th |publisher= Pearson }} Contains a lucid proof in case the measure ''ν'' is not σ-finite.
* {{cite book |last=Shilov |first=G. E. |last2=Gurevich |first2=B. L. |year=1978 |title=Integral, Measure, and Derivative: A Unified Approach |others=Richard A. Silverman, trans. |publisher=[[Dover Publications]] |isbn=0-486-63519-8 }}
* {{Cite book| publisher = Princeton University Press| isbn = 978-0-691-11386-9| last1 = Stein| first1 = Elias M.| last2 = Shakarchi| first2 = Rami| title = Real analysis: measure theory, integration, and Hilbert spaces| location = Princeton, N.J| series = Princeton lectures in analysis| date = 2005}} Contains a proof of the generalisation.
* {{cite web |last=Teschl |first=Gerald |author-link=Gerald Teschl |title=Topics in Real and Functional Analysis |url=https://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html |others=(lecture notes) }}

{{PlanetMath attribution|id=3998|title=Radon–Nikodym theorem}}

{{Measure theory}}

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