Measure Theory 14 | Radon-Nikodym theorem and Lebesgue's decomposition theorem [dark version]

Two foundational results in measure theory—Lebesgue’s decomposition theorem and the Radon–Nikodym theorem—turn complicated measures into simpler pieces: one part behaves like an ordinary density with respect to a reference measure, while the remaining part lives on a “length-zero” set. Together, they explain how any reasonable measure on the real line can be split into an absolutely continuous component and a singular component, and then (for the absolutely continuous part) rewritten as an integral against Lebesgue measure. This matters because it converts abstract measure statements into concrete formulas involving functions, which is essential in probability, analysis, and stochastic modeling.

The setup starts with a measure space (X, 𝒜, μ) and focuses on the standard case X = ℝ, 𝒜 = Borel σ-algebra 𝓑(ℝ), and μ taken relative to Lebesgue measure Λ (the “reference measure”). Lebesgue measure is characterized by assigning each interval its usual length. With Λ in place, two key notions define how a measure μ relates to Λ.

A measure μ is absolutely continuous with respect to Λ (written μ ≪ Λ) if every set that has Λ-measure zero also has μ-measure zero. In other words, μ cannot “see” any set that Lebesgue measure declares negligible. The transcript highlights simple examples: Λ is absolutely continuous with respect to itself, and the zero measure is also absolutely continuous. By contrast, μ is singular with respect to Λ (written μ ⟂ Λ) if there exists a measurable set N such that Λ(N) = 0 while μ(ℝ \ N) = 0. Intuitively, μ concentrates entirely on a Λ-null set. A canonical example is the Dirac measure at 0, denoted Δ₀: it assigns mass 1 to {0} and 0 elsewhere, making it singular relative to Lebesgue measure because {0} has Λ-measure zero.

Lebesgue’s decomposition theorem then states that, under appropriate conditions (notably σ-finiteness), any measure μ on ℝ can be uniquely decomposed into two measures on the same σ-algebra: μ = μ_AC + μ_s, where μ_AC is absolutely continuous with respect to Λ and μ_s is singular with respect to Λ. The subscripts are chosen to reflect the behavior: “AC” for absolutely continuous and “s” for singular. This uniqueness means the split is not just possible—it is determined.

The Radon–Nikodym theorem addresses the absolutely continuous part. When μ_AC ≪ Λ and the relevant σ-finiteness assumptions hold, there exists a measurable function h: ℝ → [0, ∞) (a density) such that for every measurable set A, μ_AC(A) can be written as an integral of h over A with respect to Λ: μ_AC(A) = ∫_A h dΛ. The practical payoff is that specifying μ_AC is equivalent to specifying the density function h, turning measure-theoretic complexity into an ordinary integral.

Finally, the transcript stresses that these theorems do not hold in full generality: σ-finiteness is required for both the measure μ and the reference measure (automatically satisfied for Lebesgue measure). With those conditions in place, the pair of theorems becomes a powerful toolkit, with major applications in areas like stochastic processes where measures and densities are central.