Convergence Theorems - Key Properties

The convergence theorems establish conditions under which limits and integrals can be interchanged. These results are fundamental tools for analysis, probability theory, and applied mathematics.

TheoremComparison of Convergence Modes

On a finite measure space $(X, \mathcal{F}, \mu)$ with $\mu(X) < \infty$ :

If $f_n \to f$ uniformly, then $f_n \to f$ in $L^p$ for all $p \in [1, \infty)$
If $f_n \to f$ in $L^p$ , then $f_n \to f$ in measure
If $f_n \to f$ a.e., then $f_n \to f$ in measure
If $f_n \to f$ in measure, there exists a subsequence $f_{n_k} \to f$ a.e.

The finiteness of the measure space is essential for (3) and (4).

ExampleThe Typewriter Sequence

Consider the sequence on $[0, 1]$ with Lebesgue measure defined as follows. Partition $[0,1]$ into $k$ equal intervals and let the $(k(k-1)/2 + j)$ -th function be the indicator of the $j$ -th interval in the $k$ -partition.

Explicitly: $f_1 = \chi_{[0,1]}$ , $f_2 = \chi_{[0,1/2]}$ , $f_3 = \chi_{[1/2,1]}$ , $f_4 = \chi_{[0,1/3]}$ , $f_5 = \chi_{[1/3,2/3]}$ , $f_6 = \chi_{[2/3,1]}$ , etc.

Then:

$f_n \to 0$ in measure: $\lambda(\{x : |f_n(x)| \geq \epsilon\}) = 1/k_n \to 0$
$f_n \not\to 0$ pointwise anywhere: every point is in infinitely many intervals
However, we can extract subsequences converging a.e. to $0$

Remark

Key relationships for convergence theorems:

Egorov's Theorem: On finite measure spaces, a.e. convergence is "almost" uniform convergence (uniform outside a set of small measure).
Riesz-Fischer Theorem: $L^p$ spaces are complete: Cauchy sequences in $L^p$ converge to an $L^p$ function. Moreover, there exists a subsequence converging a.e.
Vitali Convergence Theorem: On finite measure spaces, $f_n \to f$ in $L^1$ if and only if:
- $f_n \to f$ in measure
- The sequence is uniformly integrable: for all $\epsilon > 0$ , there exists $\delta$ such that $\int_E |f_n| < \epsilon$ whenever $\mu(E) < \delta$
- The sequence is tight: there exists $E$ with $\mu(E^c) < \epsilon$ and $\int_{E^c} |f_n| < \epsilon$

Understanding these relationships helps determine which convergence theorem to apply in specific situations and what additional hypotheses may be needed.