Radon–Nikodym theorem

Radon–Nikodym theorem: Let $(X,\mathcal{A},\mu)$ be a measure space such that $\mu$ is $\sigma$-finite, and let $\nu$ be a $\sigma$-finite measure on $(X,\mathcal{A})$ that is absolutely continuous with respect to $\mu$ (written $\nu \ll \mu$, meaning $\mu(E)=0 \implies \nu(E)=0$ for all $E\in\mathcal{A}$). Then there exists a measurable function $f:X\to[0,\infty]$ such that

Moreover, $f$ is unique up to $\mu$-almost everywhere equality, and it is denoted by $\frac{d\nu}{d\mu}$ (the Radon–Nikodym derivative of $\nu$ with respect to $\mu$).

In probability, applying this to probability measures yields the notion of a “density” or likelihood ratio: if $Q \ll P$ on a probability space, then $L=\frac{dQ}{dP}$ satisfies $Q(E)=\int_E L\,dP$, and for suitable $g$ one has $\mathbb{E}_Q[g]=\mathbb{E}_P[gL]$, 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.