Radon-Nikodym theorem
In
mathematics, the
Radon-Nikodym theorem is a result in
functional analysis that states that if a
measure is
absolutely continuous with respect to another
sigma-finite measure then there is a
measurable function f, taking values in [0,∞], on the underlying space such that
The function 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 probability density of one measure with respect to another. It follows trivially from the definition of the derivative that
where is the
expectation operator.
The theorem is named for Johann Radon, who proved the theorem for the special case where the underlying space is in 1913, and for Otto Nikodym who proved the general case in 1930.