Radon–Nikodym derivatives

For positive measures ν and μ on the same measurable space, ν is absolutely continuous with respect to μ if every μ-null measurable set is also ν-null. The direction matters. Under the sigma-finite assumptions on both measures used here, the Radon–Nikodym theorem supplies a nonnegative measurable density h such that ν(A)=∫A h dμ for every measurable A. The density is unique only up to μ-null changes. It is a derivative of measures, not necessarily the ordinary derivative of a pointwise function.

The theorem's conclusion explains why the null-set condition is necessary: integrating any nonnegative measurable h over a μ-null set gives zero. Existence is the difficult converse. Sigma-finiteness is part of the standard theorem being invoked, not a decorative simplification; without appropriate hypotheses a general absolute-continuity claim does not automatically yield a density. For finite signed ν, one applies the theorem to its Jordan components when its variation is absolutely continuous with respect to a sigma-finite positive μ.

Once the identity holds for indicator functions, linearity extends it to simple functions and monotone convergence extends it to nonnegative measurable functions. Thus integrating f against ν is the same as integrating f times h against μ. For signed f, require integrability with respect to ν, equivalently finite ∫|f|h dμ. This is the measure-theoretic basis of importance weighting. It does not require a change of coordinates, and a density ratio should not be confused with a Jacobian unless a separate coordinate transformation establishes that interpretation.

For sigma-finite positive measures satisfying ν≪μ≪λ, the chain rule identifies dν/dλ with the product (dν/dμ)(dμ/dλ), interpreted with the usual almost-everywhere qualifications. A reciprocal rule requires mutual absolute continuity: then a positive finite density has reciprocal as the reverse derivative on the relevant full-measure set. If the first density vanishes on a region of positive μ measure, the reverse absolute continuity fails there. Blindly taking reciprocals can therefore lose exactly the region the weighting scheme cannot see.

A measure ν is singular with respect to μ if it is carried by a μ-null measurable set. A point mass relative to Lebesgue measure is the simplest example. In the finite positive-measure setting with a sigma-finite reference μ, the Lebesgue decomposition writes ν uniquely as an absolutely continuous part plus a singular part. The first has a Radon–Nikodym density; the second cannot be represented by any ordinary μ-density. A narrow tall density can approximate a point mass weakly, but approximation does not turn the limiting atom into a Lebesgue density.

This distinction is practical when modeling mixed data. A response distribution might have a continuous spread and a separate probability of an exact zero. Writing only a continuous density omits the atom, even if that density integrates correctly over positive values. One can instead use a reference measure containing both Lebesgue measure and the relevant point mass, then describe everything with one density relative to that reference. The choice of measure changes the representation while leaving the underlying event probabilities unchanged.