Product Measures and Fubini - Applications

TheoremConvolution Theorem

Let $f, g \in L^1(\mathbb{R}^n)$ . The convolution of $f$ and $g$ is defined by: $(f * g)(x) = \int_{\mathbb{R}^n} f(x - y) g(y) \, dy = \int_{\mathbb{R}^n} f(y) g(x - y) \, dy$

Then $f * g \in L^1(\mathbb{R}^n)$ and: $\|f * g\|_1 \leq \|f\|_1 \|g\|_1$

Moreover, convolution is commutative: $f * g = g * f$ , and associative: $(f * g) * h = f * (g * h)$ .

The proof uses Fubini's theorem to interchange the order of integration, showing that convolution is a well-defined operation on $L^1$ that makes it a Banach algebra.

ExampleSmoothing by Convolution

Let $\phi_\epsilon(x) = \frac{1}{\epsilon^n} \phi(\frac{x}{\epsilon})$ where $\phi$ is a smooth, compactly supported function with $\int \phi = 1$ (called an approximation to the identity).

For $f \in L^1(\mathbb{R}^n)$ , the convolution $f * \phi_\epsilon$ is smooth (infinitely differentiable), and: $\lim_{\epsilon \to 0} \|f * \phi_\epsilon - f\|_1 = 0$

This shows that smooth functions are dense in $L^1$ , a fundamental result for PDE theory and harmonic analysis.

Remark

Applications of product measures and Fubini:

Probability: For independent random variables $X$ and $Y$ with joint distribution $\mu \otimes \nu$ : $\mathbb{E}[h(X, Y)] = \int_{X \times Y} h \, d(\mu \otimes \nu) = \int_X \left(\int_Y h(x, y) \, d\nu(y)\right) d\mu(x)$
Fourier transform: The Fourier inversion formula and Plancherel theorem use Fubini to evaluate integrals like: $\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(x) \hat{g}(\xi) e^{2\pi i x \cdot \xi} \, d\xi \, dx$
Change of variables: For measurable $T: X \times Y \to U \times V$ , computing integrals after transformation often requires Fubini to decompose the integral.
Stochastic processes: The Kolmogorov extension theorem uses product measures on infinite products to construct processes with prescribed finite-dimensional distributions.
Differential geometry: Integration on manifolds uses Fubini to reduce integrals to coordinate patches.

The Convolution Theorem and its applications demonstrate how product measures and Fubini's theorem are central tools in modern analysis, connecting integration theory with functional analysis and harmonic analysis.