Convergence Theorems - Examples and Constructions

Practical application of convergence theorems requires understanding when each theorem applies and recognizing failure modes. We explore examples that illustrate the necessity of various hypotheses.

ExampleWhen Dominated Convergence Fails

Consider $f_n(x) = n \chi_{(0, 1/n)}(x)$ on $[0, 1]$ with Lebesgue measure.

$f_n(x) \to 0$ for all $x > 0$ (pointwise a.e.)
$\int_0^1 f_n \, d\lambda = n \cdot \frac{1}{n} = 1$ for all $n$
But $\int_0^1 0 \, d\lambda = 0 \neq 1$

The failure occurs because there is no integrable dominating function: $|f_n| = n$ on $[0, 1/n]$ grows without bound. This violates the domination hypothesis of DCT.

ExampleWhen Monotone Convergence Applies

Consider computing $\int_0^\infty \sum_{n=1}^{\infty} \frac{e^{-nx}}{n^2} \, dx$ .

Define $f_N(x) = \sum_{n=1}^{N} \frac{e^{-nx}}{n^2}$ . Then $f_N \uparrow f$ where $f(x) = \sum_{n=1}^{\infty} \frac{e^{-nx}}{n^2}$ .

By MCT: $\int_0^\infty f \, dx = \lim_{N \to \infty} \int_0^\infty f_N \, dx = \lim_{N \to \infty} \sum_{n=1}^{N} \frac{1}{n^2} \int_0^\infty e^{-nx} \, dx = \sum_{n=1}^{\infty} \frac{1}{n^3} = \zeta(3)$

where we used $\int_0^\infty e^{-nx} \, dx = 1/n$ .

Remark

Practical strategies for applying convergence theorems:

Check monotonicity first: If the sequence is monotone and non-negative, use MCT. It's the simplest and most powerful result.
Look for dominating functions: For DCT, common dominating functions include:
- Bounded functions on finite measure spaces
- Functions like $e^{-cx}$ on $(0, \infty)$
- Polynomial bounds combined with exponential decay
Use Fatou's Lemma for one-sided estimates: When you only need an inequality, Fatou's Lemma is often sufficient and requires fewer hypotheses.
On finite measure spaces, use Egorov: If you have a.e. convergence on a finite measure space, Egorov's Theorem can help establish uniform convergence on large subsets.
For $L^p$ convergence: Check if $\|f_n - f\|_p \to 0$ . This is stronger than a.e. convergence but weaker than uniform convergence.

These examples demonstrate that each hypothesis in the convergence theorems serves a purpose, and removing any condition can lead to failure of the conclusion.