Convergent sequences of functions

Overview §

Convergence of sequences of functions is defined by considering the behavior of each sequence of numbers $(f_n(x))$, where $x\in X$ is any point. If the numbers $(f_n(x))$ all converge to some $f(x)$—meaning that for some arbitrary distance $ϵ>0$ we can find a sufficiently large index $N\in \mathbb{N}$ that guarantees $f_n$ and $f$ will be less than $ϵ$ apart—then $f_n\to f$ pointwise. If there is some index $N$ that shows pointwise convergence at all $x\in X$, then $f_n\to f$ uniformly.

Uniform convergence preserves both boundedness and continuity, meaning that if every function in the sequence has some property, then the uniform limit also has the property.

Relevant theorems:

• (Theorem) The limit of a uniformly convergent sequence of continuous functions is continuous

See also: Properties of uniform convergence of functions Bounded sequences of functions and equicontinuity

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Convergence of measurable functions §

Definition: Convergence for measurable functions

Let $f:E\to \mathbb{R}$ be a measurable function and let $f_n:E\to \mathbb{R}\cup \{\pm \infty \}$ be any sequence of functions to the extended real numbers.

• $f_n\to f$ pointwise if for all $x\in E$ and all $ϵ>0$, there exists $N\in N$ such that

n > N \implies |f_n(x) - f(x)| < \epsilon.

That is, the sequence of numbers $\{ f_n(x)\}$ converges to some $f(x)$ for all $x \in E$. - $f_n \to f$ **almost everywhere** if the set of all points which do not converge to $f$ has zero [[Lebesgue measure|measure]].

m({x \in E \ | \ f_n(x) \not \to f(x) }) = 0

- $f_n \to f$ **uniformly** if for all $\epsilon > 0$, there exists $N \in \mathbb N$ such that for all $x \in E$,

n > N \implies |f_n(x) - f(x) | < \epsilon.

Definition: L^p (integral) convergence

Let $f_n:X\to \mathbb{R}$ be a sequence of functions. We say $f_n\to f$ has $L^p$ convergence for a fixed $p$ if for all $ϵ>0$, there exists $N\in \mathbb{N}$ such that for all $n\ge N$ we have

$$\int |f_n(x)-f(x){|}^p<ϵ.$$

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In metrizable (topological) spaces §

Uniformity of convergence for a sequence on functions depends on both the topology of their range and the metric.

Definition: Uniform convergence in metrizable spaces

Let $f_n:X\to Y$ be a sequence of functions from the set $X$ to the metric space $Y$. We say $(f_n)$ converges uniformly to $f:X\to Y$ if for all $ϵ>0$, there exists an integer $N$ such that for all $n\ge N$ and all $x\in X$, we have

$$d(f_n(x),f(x))<ϵ.$$

Abstract Topology HW 3.5: Relationship between uniform convergence and the uniform metric

Let $X$ be a set and let $f_n:X\to \mathbb{R}$ be a sequence of functions. Let $\overline{\rho }$ be the uniform metric on the space ${\mathbb{R}}^X$ of all functions $X\to \mathbb{R}$. If $f_n:X\to \mathbb{R}$ is a sequence of functions, show that $(f_n)$ converges uniformly to the function $f:X\to \mathbb{R}$ if and only if the sequence $(f_n)$ converges to $f$ as elements of the metric space $({\mathbb{R}}^X,\overline{\rho })$.