To state the Radon-Nikodym theorem, we first need a definition. Let (X
) be a measure
space. Given two measures
ν and μ on (X
), ν is said to be absolutely continuous with respect to μ if ν(A
) = 0 for every A
for which μ(A
) = 0.
Now that that is out of the way, here is the statement of the theorem.
Let (X, B, μ) be a σ-finite measure space, and ν a measure defined on B that is absolutely continuous with respect to μ. Then there exists a nonnegative measurable function such that for each E in B
ν(E) = ∫E dμ
If g is any other function with this property, then g = almost everywhere with respect to the measure μ
The function is called the Radon-Nikodym derivative of ν with respect to μ and is denoted by dν/dμ.
Note: The integral of a function : X → R that is measurable with respect to a measure μ is defined in the same way as the Lebesgue integral. Simply replace m with μ throughout.