Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.

Definition

Suppose $(f_{n})$ is a sequence of functions sharing the same domain and codomain. The codomain is most commonly the reals, but in general can be any metric space. The sequence $(f_{n})$ converges pointwise to the function $f$, often written as

$\lim _{n\rightarrow \infty }f_{n}=f\ {\mbox{pointwise}},$

if and only if

$\lim _{n\rightarrow \infty }f_{n}(x)=f(x)$

for every x in the domain. The function $f$ is said to be the pointwise limit function of $f_{n}$.

Properties

This concept is often contrasted with uniform convergence. To say that

$\lim _{n\rightarrow \infty }f_{n}=f\ {\mbox{uniformly}}$

means that

$\lim _{n\rightarrow \infty }\,\sup\{\,\left|f_{n}(x)-f(x)\right|:x\in A\,\}=0,$

where $A$ is the common domain of $f$ and $f_{n}$. That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent. For example, if $f_{n}:[0,1)\rightarrow [0,1)$ is a sequence of functions defined by $f_{n}(x)=x^{n}$, then $\lim _{n\rightarrow \infty }f_{n}(x)=0$ pointwise on the interval [0,1), but not uniformly.

The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example,

$f(x)=\lim _{n\rightarrow \infty }\cos(\pi x)^{2n}$

takes the value 1 when x is an integer and 0 when x is not an integer, and so is discontinuous at every integer.

The values of the functions f_n need not be real numbers, but may be in any topological space, in order that the concept of pointwise convergence make sense. Uniform convergence, on the other hand, does not make sense for functions taking values in topological spaces generally, but makes sense for functions taking values in metric spaces, and, more generally, in uniform spaces.

Topology

Pointwise convergence is the same as convergence in the product topology on the space Y^X, where X is the domain and Y is the codomain. If the codomain Y is compact, then, by Tychonoff's theorem, the space Y^X is also compact.

Almost everywhere convergence

In measure theory, one talks about almost everywhere convergence of a sequence of measurable functions defined on a measurable space. That means pointwise convergence almost everywhere, i.e. on a subset of the domain whose complement has measure zero. Egorov's theorem states that pointwise convergence almost everywhere on a set of finite measure implies uniform convergence on a slightly smaller set.

Almost everywhere pointwise convergence on the space of functions on a measure space does not define the structure of a topology on the space of measurable functions on a measure space (although it is a convergence structure). For in a topological space, when every subsequence of a sequence has itself a subsequence with the same subsequential limit, the sequence itself must converge to that limit.

But consider the sequence of so-called "galloping rectangles" functions. Let N = Floor(log₂ n) and k = n mod 2^N. And let

$f_{n}(x)={\begin{cases}1,&{\frac {k}{2^{N}}}\leq x\leq {\frac {k+1}{2^{N}}}\\0,&{\text{otherwise}}.\end{cases}}.$

Then any subsequence of the sequence {f_n}_n has a sub-subsequence which itself converges almost everywhere to zero, for example by taking the subsequence of functions which do not vanish at x = 0. But at no point does the sequence converge pointwise to zero. Hence, unlike convergence in measure and L^p convergence which do induce topologies, pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.

See also

• Modes of convergence (annotated index)