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.