Lebesgue Integration - Examples and Constructions

Computing Lebesgue integrals directly from the definition can be challenging. We explore several techniques and examples that illustrate the power and flexibility of the Lebesgue integral.

ExampleRelationship with Riemann Integration

For bounded functions on intervals, there is a clear relationship:

Theorem: If $f: [a, b] \to \mathbb{R}$ is Riemann integrable, then $f$ is Lebesgue measurable and Lebesgue integrable, with: $\int_{[a,b]} f \, d\lambda = \int_a^b f(x) \, dx \quad \text{(Riemann)}$

The converse is false: the function $f = \chi_{\mathbb{Q} \cap [0,1]}$ is Lebesgue integrable with $\int_0^1 f \, d\lambda = 0$ (since $\lambda(\mathbb{Q}) = 0$ ), but is not Riemann integrable.

ExampleComputing Specific Integrals

Dirichlet function: Let $f = \chi_{\mathbb{Q}}$ on $[0, 1]$ . Then: $\int_0^1 f \, d\lambda = \lambda(\mathbb{Q} \cap [0,1]) = 0$
Power functions: For $p > -1$ : $\int_0^1 x^p \, d\lambda = \frac{1}{p+1}$

This can be verified by approximating with step functions or using the relationship with Riemann integration.

Exponential on $\mathbb{R}^+$ : $\int_0^\infty e^{-x} \, d\lambda = 1$
Gaussian integral: $\int_{-\infty}^\infty e^{-x^2} \, d\lambda = \sqrt{\pi}$

Remark

Useful techniques for computing Lebesgue integrals include:

Layer cake representation: For non-negative $f$ : $\int_X f \, d\mu = \int_0^\infty \mu(\{x : f(x) > t\}) \, dt$

This transforms integration over the domain into integration over the range.

Change of variables: If $\phi: X \to Y$ is measurable and invertible, and $f$ is non-negative or integrable: $\int_Y f(y) \, d\nu(y) = \int_X f(\phi(x)) \frac{d\nu}{d\mu}(x) \, d\mu(x)$

where $d\nu/d\mu$ is the Radon-Nikodym derivative (when it exists).

Fubini's theorem: For product measures (discussed in Chapter 7): $\int_{X \times Y} f(x, y) \, d(\mu \times \nu) = \int_X \left(\int_Y f(x, y) \, d\nu(y)\right) d\mu(x)$

These computational tools, combined with the powerful convergence theorems, make the Lebesgue integral a versatile framework for analysis, probability, and applied mathematics.