Measure Theory
Regular Measures and Riesz Representation Theorem
The Riesz theorem explains why probability measures and distribution functions are the same thing. Every way of linearly computing averages of functions is integration with respect to some measure. This is the foundation for understanding duality in functional analysis and optimal transport.
- Optimal transport theory: Wasserstein distance via regular measures
- Numerical methods: quadrature rules (weighted sums) ↔ discrete measures (Riesz theorem)
- Bayesian statistics: priors as regular measures on parameter spaces
- GANs (Generative Adversarial Networks): discriminator evaluates a Riesz functional
Regularity of Measures: Approximation by Open and Compact Sets
Radon measures underpin ML: the Wasserstein distance in Wasserstein GAN (2017) uses σ-finite Borel measures on compact spaces. A Borel measure μ on a locally compact Hausdorff space X is called **regular** if it is simultaneously **outer regular** (μ(E) = inf{μ(U) : U ⊇ E open}) and **inner regular** (μ(E) = sup{μ(K) : K ⊆ E compact}). Regularity means: the measure can be approximated arbitrarily closely by open or compact sets.
**Regularity on ℝⁿ:** Lebesgue measure λⁿ is regular on ℝⁿ: - Outer: for any E and ε>0, there exists an open U ⊇ E with λⁿ(U) < λⁿ(E) + ε - Inner: for any E and ε>0, there exists a compact K ⊆ E with λⁿ(K) > λⁿ(E) - ε **Radon measure** = locally finite, inner regular Borel measure (the standard class in geometric analysis). **Dirac measure δₓ** is regular: δₓ(E) = 1 if x∈E, 0 otherwise. Inner regularity: K = {x} ⊆ E when x∈E - a compact set with δₓ(K) = 1.
What does it mean for a measure μ to be inner regular?
Riesz Representation Theorem: Functionals ↔ Measures
**Riesz Representation Theorem:** Let X be a locally compact Hausdorff space and C₀(X) the space of continuous functions vanishing at infinity. Then for any **positive linear functional** Λ: C₀(X) → ℝ (Λ(f) ≥ 0 whenever f ≥ 0), there exists a unique regular Borel measure μ such that Λ(f) = ∫f dμ for all f ∈ C₀(X).
**The meaning of the Riesz theorem:** 'Every way of linearly evaluating functions is integration with respect to some measure.' Examples of functionals: - Λ(f) = f(x₀) → Dirac measure δₓ₀ - Λ(f) = ∫₀¹ f(x)dx → Lebesgue measure on [0,1] - Λ(f) = Σ cₙ·f(xₙ) → discrete measure Σ cₙ·δₓₙ - Λ(f) = ∫₀¹ f(x)·w(x)dx → measure with density w **Duality:** (C₀(X))* ≅ M(X) - the space of finite regular Borel measures. This is a key statement of functional analysis linking 'classical' and 'measure-theoretic' integration.
Which functional Λ: C₀(ℝ) → ℝ corresponds to the Dirac measure δₐ by the Riesz theorem?
Lebesgue-Stieltjes Integral and Measures on ℝ
The **Lebesgue-Stieltjes integral** ∫f dF generalizes the Riemann-Stieltjes integral: for any non-decreasing right-continuous function F: ℝ → ℝ, there exists a unique Borel measure μ_F with μ_F((a,b]) = F(b) - F(a). Conversely, every finite Borel measure on ℝ is determined by such a function F (a distribution function).
**Correspondence between functions and measures on ℝ:**
| Type of F | Measure μ_F | Example |
|---|---|---|
| F(x) = x (continuous) | Lebesgue measure λ | ∫f dx |
| F(x) = 1_{x≥0} (step) | Dirac δ₀ | f(0) |
| F(x) = (1-e^{-x})·1_{x≥0} | Exponential distribution | ∫₀^∞ f e^{-x}dx |
| F = CDF of random var. X | Distribution P_X | E[f(X)] |
| F = Cantor function (continuous but constant a.e.) | Cantor measure (⊥ Lebesgue) | - |
**Riesz + Stieltjes:** Every finite regular measure on ℝ ↔ distribution function (right-continuous, monotone). This is the bridge between 'probabilistic' and 'measure-theoretic' language.
Which measure μ_F corresponds to F(x) = 1_{x ≥ 0} (a step at zero) via the Lebesgue-Stieltjes integral?
Key Ideas
- Regular measure: approximable from outside (by open sets) and inside (by compact sets)
- Riesz theorem: Λ positive linear functional on C₀(X) ↔ ∃! regular measure μ with Λ(f) = ∫f dμ
- Λ(f) = f(a) ↔ δₐ; Λ(f) = ∫f dx ↔ Lebesgue measure
- (C₀(X))* ≅ M(X) - the space of regular measures
- Lebesgue-Stieltjes integral: ∫f dF, F - distribution function ↔ Borel measure
- Cantor measure: example of a regular measure singular with respect to Lebesgue
Related Topics
The Riesz theorem is the meeting point of measure theory and functional analysis:
- Radon-Nikodym Theorem — R-N: absolutely continuous measures have densities; Riesz: all measures = functionals
- Ergodic Theory — Invariant measures are regular measures with additional structure
Вопросы для размышления
- How is the Riesz theorem related to quadrature formulas in numerical integration?
- Can weak convergence of measures μₙ → μ be expressed via the Riesz theorem? (Hint: it is convergence as functionals on C₀(X).)
- How does the Cantor measure differ from Lebesgue measure and Dirac measure? On which set does it 'live'?