Probability with Measure

2.3 Convergence of functions

We’ve thought about convergence of (extended) real numbers in Sections 2.1 and 2.2. In this section we will think about convergence of functions, defined on a general measure space $(S, Σ, m)$ . We will write this section for functions taking values in $R$ , but it applies equally to functions taking values in $\overset{―}{R}$ .

First we need to introduce pointwise definitions of functions. This is best done by example. If $f$ and $g$ are functions defined from $S \to R$ then we define the function $f + g : S \to R$ by setting $(f + g) (x) = f (x) + g (x)$ . We can apply the same idea to $f g$ to define the function $(f g) (x) = f (x) g (x)$ , and so on.

Definition 2.3.1 Let $f_{n}$ and $f$ be functions defined from $: S \to R$ . We say that $f_{n} \to f$ pointwise if $f_{n} (x) \to f (x)$ as $n \to \infty$ , for all $x \in S$ .

Pointwise convergence is the simplest type of convergence of functions. You’ve already seen one other type: if $sup_{x \in A} | f_{n} (x) - f (x) | \to 0$ as $n \to \infty$ , then we say that $f_{n} \to f$ uniformly on the set $A$ . We will use pointwise and uniform convergence within this course, but for us the most interesting type of convergence is something slightly different. Recall the term ‘almost all’ from Section 1.8. A property holds for almost all $x \in S$ if the set of $x$ on which it fails is a null set.

Definition 2.3.2 Let $f_{n}$ and $f$ be functions defined from $: S \to R$ . We say that $f_{n} \to f$ almost everywhere if $f_{n} (x) \to f (x)$ for almost all $x \in S$ .

We will sometimes abbreviate $f_{n} \to f$ almost everywhere as $f_{n} \overset{a . e .}{\to} f$ . Unpacking the terminology in Definition 2.3.2, we have that $f_{n} \overset{a . e .}{\to} f$ if and only $m ({x \in S; f_{n} (x) ↛ f (x)}) = 0$ .

Convergence almost everywhere is very similar to pointwise convergence. The difference is that we allow $f_{n} (x) \to f (x)$ to fail on some null set of $x \in S$ . This is much more natural from the perspective of measure theory, because we want to forget about things that have measure zero.

Example 2.3.3 Let $f_{n} : R \to R$ by $f_{n} (x) = e^{- n x^{2}}$ and let $f (x) = 0$ . We take our measure space to be $(R, B (R), λ)$ , and note that as $n \to \infty$ we have $f_{n} (x) \to 0$ for all $x \in R$ except $x = 0$ (at which $f_{n} (0) = 1$ ). The set ${0}$ is Lebesgue null, so $f_{n} \overset{a . e .}{\to} f$ .

Lemma 2.3.4 Let $f_{n}$ and $f$ be functions from $S \to R$ .
- 1. If $f_{n} \to f$ uniformly then $f_{n} \to f$ pointwise.
- 2. If $f_{n} \to f$ pointwise then $f_{n} \to f$ almost everywhere.

Proof: For the first claim, if $sup_{x \in A} | f_{n} (x) - f (x) | \to 0$ then, for any $x \in A$ , we have $f_{n} (x) \to f (x)$ . For the second claim, pointwise convergence implies that the set ${x \in S; f_{n} (x) ↛ f (x)}$ is empty, hence it has measure zero. ∎

Example 2.3.3 shows that we can have convergence almost everywhere without having pointwise convergence. Exercise 2.7 gives an example of functions that converge pointwise but not uniformly.