Radon–Nikodym theorem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Radon–Nikodym theorem is a result in <a href="/facts/Measure_theory/yCq7zlI4">measure theory</a> that expresses the relationship between two measures defined on the same <a href="/facts/Measurable_space/AJ98lhT4">measurable space</a>. A measure is a <a href="/facts/Set_function/Dy46B9EJ">set function</a> 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 <a href="/facts/Probability_space/dEcv2VrU">probability space</a>.
One way to derive a new measure from one already given is to assign a density to each point of the space, then <a href="/facts/Lebesgue_integration/f4AgvC6x">integrate</a> over the measurable subset of interest. This can be expressed as

ν
        (
        A
        )
        =
        
          ∫
          
            A
          
        
        f
        
        d
        μ
        ,
      
    
    {\displaystyle \nu (A)=\int _{A}f\,d\mu ,}

where ν is the new measure being defined for any measurable subset A and the function f is the density at a given point. The integral is with respect to an existing measure μ, which may often be the canonical <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a> on the <a href="/facts/Real_line/6eq42xX5">real line</a> R or the n-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> Rn (corresponding to our standard notions of length, area and volume). For example, if f represented mass density and μ was the Lebesgue measure in three-dimensional space R3, then ν(A) would equal the total mass in a spatial region A.
The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ν can be expressed in this way with respect to another measure μ on the same space. The function  f  is then called the Radon–Nikodym derivative and is denoted by 
 
 
 
 
 
 
 
 d
 ν
 
 
 d
 μ
 
 
 
 
 
 
 {\displaystyle {\tfrac {d\nu }{d\mu }}}
 
. An important application is in <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, leading to the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> of a <a href="/facts/Random_variable/TwTBXnLT">random variable</a>.
The theorem is named after <a href="/facts/Johann_Radon/sJxXdkDo">Johann Radon</a>, who proved the theorem for the special case where the underlying space is Rn in 1913, and for <a href="/facts/Otto_Nikodym/JfBYgABM">Otto Nikodym</a> who proved the general case in 1930. In 1936 <a href="/facts/Hans_Freudenthal/l8V4uA0N">Hans Freudenthal</a> generalized the Radon–Nikodym theorem by proving the <a href="/facts/Freudenthal_spectral_theorem/t9oFUqsN">Freudenthal spectral theorem</a>, a result in <a href="/facts/Riesz_space/FX7Mwfxe">Riesz space</a> theory; this contains the Radon–Nikodym theorem as a special case.
A <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> Y is said to have the <a href="/facts/Radon%E2%80%93Nikodym_property/RpzcTu9n">Radon–Nikodym property</a> if the generalization of the Radon–Nikodym theorem also holds, <a href="/facts/Mutatis_mutandis/okly010I">mutatis mutandis</a>, for functions with values in Y. All <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a> have the Radon–Nikodym property.

Radon–Nikodym theorem open-in-new

Radon–Nikodym theorem