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

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 /.

Note: The integral of a function ƒ : XR 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.

Log in or register to write something here or to contact authors.