Densities and Radon-Nikodym derivatives are the measure-theoretic language of changing reference measure. They unify likelihood ratios, change of variables, and change of probability.
Absolute continuity and densities
A density is a derivative of one measure with respect to another.
Definition. For measures \(\nu,\mu\) on \((S,\mathcal S)\), write \(\nu\ll\mu\) if \(\mu(A)=0\) implies \(\nu(A)=0\). A Radon-Nikodym derivative \(\mathrm{d}\nu/\mathrm{d}\mu\) is a nonnegative measurable function \(f\) such that \(\nu(A)=\int_A f\,\mathrm{d}\mu\) for every \(A\in\mathcal S\).
Radon-Nikodym theorem
The theorem is the rigorous existence result behind densities.
Theorem. If \(\nu\ll\mu\) and both measures are \(\sigma\)-finite, then there exists a measurable \(f\ge0\), unique \(\mu\)-a.e., such that \(\nu(A)=\int_A f\,\mathrm{d}\mu\) for all \(A\).
Proof. A complete Hilbert-space proof is given in Real analysis. In brief, for finite measures set \(\rho=\mu+\nu\). Riesz representation on \(L^2(\rho)\) gives \(h\in[0,1]\) with \[ \nu=h\rho,\qquad \mu=(1-h)\rho. \] Absolute continuity forces \(\rho(h=1)=0\), so \(f=h/(1-h)\) satisfies \(\nu=f\mu\). Sigma-finite pieces are then pasted, and uniqueness follows by integrating over \(\{f>g\}\) and \(\{g>f\}\).
Proposition (chain and reciprocal rules). If \(R\ll Q\ll P\), then \[ \frac{\mathrm{d}R}{\mathrm{d}P} =\frac{\mathrm{d}R}{\mathrm{d}Q}\frac{\mathrm{d}Q}{\mathrm{d}P} \quad P\text{-a.s.} \] If \(P\) and \(Q\) are equivalent, then \[ \frac{\mathrm{d}P}{\mathrm{d}Q} =\left(\frac{\mathrm{d}Q}{\mathrm{d}P}\right)^{-1} \quad Q\text{-a.s.} \]
Proof. Substitute the derivatives into integrals: \[ R(A)=\int_A\frac{\mathrm{d}R}{\mathrm{d}Q}\,\mathrm{d}Q =\int_A\frac{\mathrm{d}R}{\mathrm{d}Q}\frac{\mathrm{d}Q}{\mathrm{d}P}\,\mathrm{d}P. \] Uniqueness gives the chain rule. Apply it to \(P\ll Q\ll P\) to obtain the reciprocal identity.
Change of probability
Likelihood ratios reweight expectations.
Proposition. If \(Q\ll P\) with \(Z=\mathrm{d}Q/\mathrm{d}P\), then for every nonnegative or \(Q\)-integrable \(X\), \[ \mathbb E_QX=\mathbb E_P[ZX]. \]
Proof. For \(X=1_A\), the identity is \(Q(A)=\int_AZ\,\mathrm{d}P\). Extend to nonnegative simple functions by linearity, to nonnegative measurable functions by monotone convergence, and to integrable functions by decomposing into positive and negative parts.
Corollary (Bayes formula between two laws). If \(P\) and \(Q\) are equivalent and \(Z=\mathrm{d}Q/\mathrm{d}P\), then for a sub-sigma-field \(\mathcal G\) and \(Q\)-integrable \(X\), \[ \mathbb E_Q[X\mid\mathcal G] =\frac{\mathbb E_P[ZX\mid\mathcal G]} {\mathbb E_P[Z\mid\mathcal G]} \quad Q\text{-a.s.}, \] with the ratio interpreted on the set where the denominator is positive.
Proof. The right side is \(\mathcal G\)-measurable. For \(G\in\mathcal G\), \[ \mathbb E_Q\!\left[ \mathbf1_G\frac{\mathbb E_P[ZX\mid\mathcal G]} {\mathbb E_P[Z\mid\mathcal G]} \right] =\mathbb E_P\!\left[ Z\mathbf1_G\frac{\mathbb E_P[ZX\mid\mathcal G]} {\mathbb E_P[Z\mid\mathcal G]} \right]. \] Conditioning the factor \(Z\) on \(\mathcal G\) cancels the denominator, leaving \(\mathbb E_P[\mathbf1_GZX]=\mathbb E_Q[\mathbf1_GX]\). This is the defining identity.
Densities under smooth transformations
Theorem. Let \(X\) have density \(p_X\) on an open \(U\subset\mathbb R^d\), and let \(T:U\to V\) be a \(C^1\) diffeomorphism. Then \(Y=T(X)\) has density \[ p_Y(y) =p_X(T^{-1}(y)) \left|\det DT^{-1}(y)\right|. \]
Proof. For bounded measurable \(g\), \[ \mathbb Eg(Y) =\int_Ug(T(x))p_X(x)\,\mathrm{d}x. \] The multivariable change-of-variables theorem with \(y=T(x)\) turns this into \[ \int_Vg(y)p_X(T^{-1}(y)) |\det DT^{-1}(y)|\,\mathrm{d}y. \] Uniqueness of densities gives the formula.
Example (lognormal law). If \(X\sim N(\mu,\sigma^2)\) and \(Y=e^X\), then \(T^{-1}(y)=\log y\) and \(|(T^{-1})'(y)|=1/y\). Hence \[ p_Y(y)=\frac1{y\sigma\sqrt{2\pi}} \exp\!\left[-\frac{(\log y-\mu)^2}{2\sigma^2}\right], \qquad y>0. \]
Remark. In statistics, \(Z\) is a likelihood ratio; in stochastic calculus, carefully chosen \(Z_t\) changes drift while preserving martingale structure under the new law.