Radon–Nikodym: Densities Relative to Any Measure
The course's last pillar: when does one measure have a density with respect to another? The answer underwrites conditional expectation, closes the loop on the "zoo" of laws from Chapter 4, and produces — in the Cantor law — the honest negative example: a measure with no density in any direction.
In this subchapter
8.2.1Absolute continuity: the hypothesis
A measure ν is absolutely continuous with respect to μ (written ν ≪ μ) if every μ-null set is ν-null: μ(A) = 0 ⟹ ν(A) = 0. Equivalently (an exercise in ε-bounding worth doing): for every ε >0 there is δ >0 with μ(A) < δ ⟹ ν(A) < ε — the continuity reading that gave the concept its name.
Any law with a density f: ν(B) = ∫B f dμ — then μ(A) = 0 ⟹ ∫A f dμ = 0 (integrals over null sets vanish, 3.1): densities imply absolute continuity, trivially. The question of the theorem is the converse: does every ≪ come from a density? The answer is yes — and that is the entire content: the converse is a theorem, not a definition. (For a pair of probability measures on the same σ-algebra, every absolute continuity claim is exactly a "can we reweight μ into ν" claim — the mathematics of importance sampling, likelihood ratios, and change of measure, the quant's daily bread: the Radon–Nikodym derivative is the likelihood ratio, now with an existence theorem behind it.)
8.2.2The theorem, proved in the L² case
If ν ≪ μ and ν is finite, there is a measurable f ≥ 0 (the Radon–Nikodym derivative, dν/dμ) with ν(A) = ∫A f dμ for every measurable A; f is unique up to μ-null changes. We prove the case ν ≤ μ up to a constant — specifically ν ≤ c·μ with both finite — which covers probability-conditioning applications; the general finite case follows by decomposing ν into such pieces (flagged, standard).
Assume ν ≤ μ (rescaling handles ν ≤ cμ). Consider the space L²(μ) — measurable functions with ∫g² dμ < ∞, identified up to μ-null equality (the a.s. license, institutionalized). Define the convex set 𝒦 = { g ∈ L²(μ) : g ≥ 0, ∫A g dμ ≤ ν(A) for all A } — the μ-densities that under-estimate ν everywhere.
Step 1 — 𝒦 is nonempty and closed under midpoint. 0 ∈ 𝒦. If g, h ∈ 𝒦, then (g+h)/2 ∈ 𝒦: split any set A into the region A ∩ {g ≥ h} — on which the average is bounded, integral-wise, by g's membership bound — and its complement, on which it is bounded by h's; summing the two bounds gives membership for the average. So 𝒦 is convex.
Step 2 — find the element of 𝒦 with maximal total mass. Let M = sup{ ∫g dμ : g ∈ 𝒦 } — finite (≤ ν(Ω) < ∞) and nonzero (unless ν = 0, trivial case). First, 𝒦 is also closed under pointwise maxima: for g, h ∈ 𝒦, split A into {g ≥ h} (bound by g) and its complement (bound by h) to see max(g,h) ∈ 𝒦 — the same two-region split as the convexity step. Now take gn ∈ 𝒦 with integrals → M and replace each by hn = max(g₁, …, gn) ∈ 𝒦: the hn increase, stay in 𝒦, and their integrals still climb to M. Let hn ↑ g*: the MCT of 5.2 gives both ∫g* dμ = M and (a second MCT run, over each fixed A) that g* ∈ 𝒦. The maximal element exists.
Step 3 — show the residual is zero. Define ρ(A) = ν(A) − ∫A g* dμ ≥ 0: a measure (difference of measures, nonnegative by membership). If ρ ≠ 0, some set has ρ(B) > 0, hence (ε-management: scale) some ε >0 with the signed measure ρ − εμ positive on some set B — then g* + ε·1B ∈ 𝒦 (its integral over any A is g*'s plus εμ(A∩B) ≤ ν(A): the two memberships compose), and its total integral exceeds M by ε μ(B) > 0 — contradiction with maximality. So ρ = 0: ν(A) = ∫A g* dμ for every A. The derivative is g*. ∎
Two steps above were flagged rather than fully expanded (the convexity bookkeeping, the lattice closure). The full arguments are ε-exercises of the kind the course has drilled since 5.1's zigzag; a reader who has followed this far can complete them, and should — once — since the Radon–Nikodym theorem is the load-bearing wall of everything in this chapter. The alternative proof via martingale convergence (a Doob theorem from 8.3 technology) is the graduate-text route; both flags point at the same structure: the theorem is deep enough to be worth two proofs in a lifetime.
8.2.3The Cantor law: singular, honestly
The Cantor law μC (4.2 §5): μC(C) = 1 for the Cantor set C, and λ(C) = 0 (3.3). Is μC ≪ λ? No: C is λ-null but μC-full — the hypothesis of the theorem fails at a single set, and no λ-density of the Cantor law exists. (Sanity: the "derivative of the staircase" computation gives 0 a.e., which would yield the zero measure, not μC — 5.4 Example 2's failed script, now with the reason.) The Cantor law is singular with respect to λ: the two measures live on disjoint sets (C and its complement). Singularity is the exact opposite of absolute continuity — one measure's full set is the other's null set — and both are legitimate probability measures. There is no "better behaved than it looks" rescue: densities are a relationship between two measures, not a property of one, and the relationship simply does not hold here.
8.2.4Lebesgue decomposition: the zoo, closed
Any finite measure ν splits uniquely as ν = νac + νsing with νac ≪ μ (hence has a density: dνac/dμ exists) and νsing ⊥ μ (singular). Combined with the discrete/continuous split of 4.2, the classification promised in 5.4 Example 1 closes: every law is a unique mixture of an atomless-density part, an atomic part, and a singular-continuous part. The zoo has exactly three species and the exhibit is complete. (Proof of the decomposition: run the Radon–Nikodym construction on ν restricted past its μ-singular part — flagged as an exercise with all tools now in hand.)
A credit loss law: an atom at 0 (no default, mass p), an atom at recovery-capped loss levels (workout outcomes), and a continuous density over the loss range. The decomposition reads it: atomic part = the atoms' sum of Diracs; the continuous part carries the density; singular-continuous = 0 for this law (but not identically zero in general — stress-test loss models with Cantor-like structure exist, more as mathematical warnings than market features). The theorem's value is that fitting such a law means fitting three separate objects with three separate tools, and conflating them (e.g. "the density" of a mixed law — there is none) is a category error the decomposition makes impossible.
8.2.5Back to conditioning: the debt paid
Retrace 8.1 Theorem 1: for X ∈ L¹ and sub-σ-algebra G, define on G the measure Q(A) = E[X·1A] (a genuine measure: countable additivity is MCT-plus-linearity, 5.2's machinery). If X ≥ 0, Q is finite (Q(Ω) = E[X] < ∞ by integrability... for general X split into ± parts, each finite). And Q ≪ P|G: if P(A) = 0 then E[X·1A] = 0 (the integral over a null set, 4.1's a.s. discipline). Radon–Nikodym delivers dQ/d(P|G) =: Z, a G-measurable function with ∫A Z dP = Q(A) = E[X 1A] for all A ∈ G — exactly properties 1–2 of the conditional expectation. Existence paid in full; the L² projection picture of 8.1 §5 is the special case where the geometry is visible.
8.2.6What the derivative means
- Likelihood ratio / change of measure: reweight outcomes by g = dν/dμ; expectations under ν are μ-expectations of g-weighted variables. This is the theorem behind importance sampling, risk-neutral pricing's change of measure, and every "tilt" in computational finance.
- Density (μ = λ): the ordinary density of 5.4 — now seen as one instance of a general relationship, not a primitive notion.
- Conditional expectation: E[X|G] = dQ/dP with Q the X-weighted measure — conditioning is a change of measure onto the information set. The two central operations of applied probability (updating and reweighting) are the same theorem.
- Information geometry: the Hilbert-space picture of 8.1 §5 generalizes — the derivative is the "coordinate" of ν in the μ-basis, and conditioning is projection in that basis.
Absolute continuity is the permission slip for a density; the Cantor law is the denial; and conditional expectation was a change of measure all along.
The static theory is complete. The final subchapter sets it in motion: filtrations from 4.3 carrying conditional expectations from this one — martingales, the mathematics of fair games and no-free-lunch; the betting-system theorem; the optional stopping theorem (with its hypothesis-flagged failures); and a closing view: where to take the subject next.