Radon–Nikodym theorem: Let (X,A,μ) be a measure space such that μ is σ-finite, and let ν be a σ-finite measure on (X,A) that is absolutely continuous with respect to μ (written ν≪μ, meaning μ(E)=0⟹ν(E)=0 for all E∈A). Then there exists a measurable function f:X→[0,∞] such that
ν(E)=∫Efdμfor all E∈A.
Moreover, f is unique up to μ-almost everywhere equality, and it is denoted by dμdν (the Radon–Nikodym derivative of ν with respect to μ).
In probability, applying this to probability measures yields the notion of a “density” or likelihood ratio: if Q≪P on a probability space, then L=dPdQ satisfies Q(E)=∫ELdP, and for suitable g one has EQ[g]=EP[gL], linking the theorem to expectation. A common structural use is that conditional expectation can be characterized as a Radon–Nikodym derivative with respect to the restriction of a probability measure to a smaller sigma-algebra.