Product Measures and Fubini - Examples and Constructions

Understanding when and how to apply Fubini and Tonelli theorems is essential for practical integration. We explore examples that illustrate both successful applications and potential pitfalls.

ExampleWhen Fubini Fails

Consider $f(x, y) = \frac{xy}{(x^2 + y^2)^2}$ on $[0, 1] \times [0, 1] \setminus \{(0, 0)\}$ .

Computing iterated integrals: $\int_0^1 \left(\int_0^1 \frac{xy}{(x^2 + y^2)^2} \, dy\right) dx = \int_0^1 \frac{x}{2(1 + x^2)} \, dx = \frac{1}{4} \ln 2$

But reversing the order: $\int_0^1 \left(\int_0^1 \frac{xy}{(x^2 + y^2)^2} \, dx\right) dy = -\frac{1}{4} \ln 2$

The iterated integrals differ! This occurs because $f$ is not integrable: the function changes sign and $|f|$ is not integrable over the square. Fubini's theorem does not apply.

ExampleSuccessful Application

To compute $\int_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, d(\lambda \otimes \lambda)$ , we use Tonelli (since the integrand is non-negative): $\int_{\mathbb{R}} \int_{\mathbb{R}} e^{-(x^2 + y^2)} \, dy \, dx = \int_{\mathbb{R}} e^{-x^2} \left(\int_{\mathbb{R}} e^{-y^2} \, dy\right) dx$

Let $I = \int_{\mathbb{R}} e^{-x^2} \, dx$ . Then: $\int_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, d\lambda^2 = I^2$

Converting to polar coordinates (which requires a change of variables formula), the left side equals $\pi$ . Thus $I = \sqrt{\pi}$ , the famous Gaussian integral.

Remark

Practical guidelines for applying Fubini and Tonelli:

Check integrability first: For general functions, verify that $\int |f| \, d(\mu \otimes \nu) < \infty$ before applying Fubini. This can often be done using Tonelli on $|f|$ .
Use Tonelli for non-negative functions: When $f \geq 0$ , Tonelli applies automatically. No integrability check is needed.
Change of variables: For transformations $T: X \times Y \to U \times V$ , the change of variables formula involves the Jacobian determinant (in Euclidean spaces).
Infinite products: Fubini extends to countably infinite products via Kolmogorov's extension theorem, which is fundamental for constructing stochastic processes.
Mixed spaces: Product measures work for discrete-continuous pairs, such as $\mathbb{N} \times \mathbb{R}$ with counting measure times Lebesgue measure. This is useful in probability (e.g., Poisson processes).

These examples demonstrate that while Fubini and Tonelli are powerful tools, they must be applied carefully, with proper attention to integrability conditions.