Pointwise convergence

Summary

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.[1][2]

Definition

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Suppose that   is a set and   is a topological space, such as the real or complex numbers or a metric space, for example. A sequence of functions   all having the same domain   and codomain   is said to converge pointwise to a given function   often written as   if (and only if) the limit of the sequence   evaluated at each point   in the domain of   is equal to  , written as   The function   is said to be the pointwise limit function of the  

The definition easily generalizes from sequences to nets  . We say   converge pointwises to  , written as   if (and only if)   is the unique accumulation point of the net   evaluated at each point   in the domain of  , written as  

Sometimes, authors use the term bounded pointwise convergence when there is a constant   such that   .[3]

Properties

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This concept is often contrasted with uniform convergence. To say that   means that   where   is the common domain of   and  , and   stands for the supremum. 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   is a sequence of functions defined by   then   pointwise on the interval   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   when   is an integer and   when   is not an integer, and so is discontinuous at every integer.

The values of the functions   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

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Let   denote the set of all functions from some given set   into some topological space   As described in the article on characterizations of the category of topological spaces, if certain conditions are met then it is possible to define a unique topology on a set in terms of which nets do and do not converge. The definition of pointwise convergence meets these conditions and so it induces a topology, called the topology of pointwise convergence, on the set   of all functions of the form   A net in   converges in this topology if and only if it converges pointwise.

The topology of pointwise convergence is the same as convergence in the product topology on the space   where   is the domain and   is the codomain. Explicitly, if   is a set of functions from some set   into some topological space   then the topology of pointwise convergence on   is equal to the subspace topology that it inherits from the product space   when   is identified as a subset of this Cartesian product via the canonical inclusion map   defined by  

If the codomain   is compact, then by Tychonoff's theorem, the space   is also compact.

Almost everywhere convergence

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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, that is, 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, which are defined using the floor function: let   and   mod   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   convergence, pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.

See also

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References

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  1. ^ Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. ISBN 0-07-054235-X.
  2. ^ Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
  3. ^ Li, Zenghu (2011). Measure-Valued Branching Markov Processes. Springer. ISBN 978-3-642-15003-6.