Product Measures and Fubini's Theorem

How do one compute ∫∫_{ℝ²} e^{-(x²+y²)} dxdy? Directly it's hard - there's no explicit antiderivative. But Fubini's theorem plus polar coordinates does it in two lines. This is exactly how one proves that the normalizing constant of the Gaussian distribution equals √(2π).

• Numerical methods: multi-dimensional integrals via iterated one-dimensional quadrature
• Probability: E[f(X,Y)] for independent X,Y equals a double integral over the product measure
• Signal processing: convolution (f*g) is computed via Fubini - an iterated integral
• Physics: computing electrostatic potentials, quantum corrections

Product of Measurable Spaces

JAX and PyTorch compute 2D convolutions via Fubini's theorem: the double integral over (x, y) equals iterated 1D integrals. Without Fubini-Tonelli, swapping the integration order in batch normalization statistics would be formally invalid. Bayesian inference over joint distributions P(w, D) also relies on this product measure structure.

**Key facts about the product σ-algebra:** - ℬ(ℝ) ⊗ ℬ(ℝ) = ℬ(ℝ²) - Borel σ-algebras are consistent with the product - If f: X×Y → ℝ is measurable w.r.t. 𝒜⊗ℬ, then sections fₓ(y) = f(x,y) and f^y(x) = f(x,y) are also measurable - The projection X×Y → X does not necessarily preserve measurability in the reverse direction (analytic sets!) - For complete measures: (𝒜⊗ℬ)^{μ⊗ν} may be strictly larger than 𝒜^μ ⊗ ℬ^ν

Constructing Product Measures: Tonelli and Fubini

**Tonelli's Theorem** (for non-negative functions): If f: X×Y → [0,+∞] is measurable and μ, ν are σ-finite measures, then x ↦ ∫f(x,y)dν(y) and y ↦ ∫f(x,y)dμ(x) are measurable, and ∫f d(μ⊗ν) = ∫(∫f(x,y)dν)dμ = ∫(∫f(x,y)dμ)dν. **Fubini's Theorem** (for integrable functions): the same, if additionally ∫|f| d(μ⊗ν) < ∞.

**Without integrability, Fubini's theorem can yield a contradiction!** Classic counterexample: f(x,y) = (x²-y²)/(x²+y²)² on [0,1]². ∫₀¹(∫₀¹ f dx)dy = π/4 ∫₀¹(∫₀¹ f dy)dx = −π/4 The order matters! Reason: ∫|f| d(λ⊗λ) = ∞ (the function is not integrable). Tonelli saves the day: first check non-negative |f|, then apply Fubini.

Iterated Integrals and Applications

Fubini's theorem turns a multi-dimensional integral into a sequence of one-dimensional ones. This is the basic tool for computing integrals in ℝⁿ: ∫_{ℝⁿ} f dλⁿ = ∫ℝ(∫ℝ...∫ℝ f(x₁,...,xₙ)dx₁...)dxₙ. Important: σ-finiteness of both measures is mandatory.

**Applications of Fubini's theorem:** 1. **Convolution formula:** (f*g)(x) = ∫f(y)g(x-y)dy - measurability proof via Fubini 2. **Cavalieri's principle:** λⁿ(E) = ∫λⁿ⁻¹(Eₓ)dx - volume = integral of section areas 3. **Probability:** E[f(X,Y)] = ∫∫f(x,y)dP_X(x)dP_Y(y) for independent X,Y 4. **Gaussian integral:** ∫_{-∞}^{∞} e^{-x²}dx = √π - proof via product and polar coordinates 5. **Laplace transform:** computing double transforms via Fubini

Key Ideas

• 𝒜⊗ℬ - the smallest σ-algebra on X×Y containing rectangles A×B
• Sections of E ∈ 𝒜⊗ℬ are measurable: Eₓ ∈ ℬ for each x
• Tonelli: f ≥ 0 ⟹ ∫f d(μ⊗ν) = ∫(∫f dν)dμ = ∫(∫f dμ)dν
• Fubini: ∫|f|d(μ⊗ν) < ∞ ⟹ same result for integrable f
• Without integrability, the order of integration CAN affect the result
• Standard workflow: Tonelli to |f| → check integrability → Fubini to f

Related Topics

Product measures are the foundation of multi-dimensional integration and probability theory:

• Lebesgue Integral — The integral over a product measure is defined via the Lebesgue integral
• Signed Measures — Next step: measures that take negative values

Вопросы для размышления

• Why does proving ∫e^{-x²}dx = √π require Fubini's theorem? Could one do without it?
• Construct an example of a function f: [0,1]² → ℝ for which both iterated integrals exist but are not equal.
• How does Fubini's theorem relate to the notion of independence of random variables in probability theory?

Связанные уроки

• calc-17-multivariable