To state the Radon-Nikodym theorem, we first need a definition. Let (
X,
B) be a
measure space. Given two
measures ν and μ on (
X,
B), ν is said to be absolutely continuous with respect to μ if ν(
A) = 0 for every
A in
B 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 ƒ such that for each E in B
ν(E) = ∫E ƒ dμ
If g is any other function with this property, then g = ƒ almost everywhere with respect to the measure μ
The function ƒ is called the Radon-Nikodym derivative of ν with respect to μ and is denoted by dν/dμ.
Note: The integral of a function ƒ : 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.