Let ( X , X , μ ) be a measure space with μ begin σ-finite.Definition of Random Variable: Let ( Y , Y ) be a measurable space. A function ξ : X → Y is a random variable if ξ − 1 ( A ) ∈ X for every A ∈ Y . We say that the random variable is a X -measurable function.Definition of Radon-Nikodym derivative: Let λ be σ-finite measure with μ ≪ λ. Then there exists a X -measurable function f : X → [ 0 , + ∞ ) denoted f = d μ d λ satisfying μ ( A ) = ∫ A f d λ ∀ A ∈ X .It seems that the Radon-Nikodym derivative is a X -measurable function so by definition it should be a random variable between ( X , X ) and ( R ≥ 0 , B ( R ≥ 0 ) ). Is this the case that every Radon-Nikodym derivative is a random variable?

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Let   (      X    ,      X    ,  μ  ) be a measure space with   μ begin   σ-finite.Definition of Random Variable: Let   (      Y    ,      Y    ) be a measurable space. A function   ξ  :      X    →      Y   is a random variable if       ξ          −      1        (  A  )  ∈      X   for every   A  ∈      Y  . We say that the random variable is a       X  -measurable function.Definition of Radon-Nikodym derivative: Let   λ be   σ-finite measure with   μ  ≪  λ. Then there exists a       X  -measurable function   f  :      X    →  [  0  ,  +  ∞  ) denoted  f  =            d      μ              d      λ      satisfying  μ  (  A  )  =      ∫    A    f  d  λ    ∀  A  ∈      X    .It seems that the Radon-Nikodym derivative is a       X  -measurable function so by definition it should be a random variable between   (      X    ,      X    ) and   (            R              ≥      0        ,      B    (            R              ≥      0        )  ). Is this the case that every Radon-Nikodym derivative is a random variable?

Braedon Rivas · Accepted Answer

The axiomatization of probability by Kolmogorov begins by giving suggestive names to familiar objects. For example, a mere &quot;measurable function&quot; is called &quot;random variable&quot;. Of course, a &quot;random variable&quot; isn&#039;t random at all (by the way, to my knowledge, the word &quot;random&quot; has formal definitions only in some areas of math, for example, the definition of Chaitin that is more or less related to the &quot;unpredictable&quot; character of randomness - none of which are directly related to probability theory). So, any map between finite sets is a random variable, if one assumes that the finite sets are given the discrete   σ-algebra. For example, the map sending each human being to its height in centimeters is a random variable. The identity map, from {4,7,56} to itself, is also a random variable.Now, probability theory, as a theory with fancy names, has proved to be very successful in modelling &quot;true random events in real life&quot;, but this is another story; to understand why is a problem of dynamical systems, physics, maybe philosophy. But if it is just thought a branch of measure theory, it is just like any other mathematical theory.

Craig Mendoza · Accepted Answer

Short answer: yes.

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