Editing Radon–Nikodym theorem (section)

{{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.