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 formula_1 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 formula_1 converges pointwise to the function formula_3, often written as
if and only if,
for every "x" in the domain. The function formula_3 is said to be the pointwise limit function of formula_7.
Properties.
This concept is often contrasted with uniform convergence. To say that
means that
where formula_10 is the common domain of formula_3 and formula_7. 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 formula_13 is a sequence of functions defined by formula_14, then formula_15 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,
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
Then any subsequence of the sequence has a sub-subsequence which itself converges almost everywhere to zero, for example, the subsequence of functions which do not vanish at . But at no point does the original sequence converge pointwise to zero. Hence, unlike convergence in measure and "L""p" convergence, pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.