Editing Pointwise convergence (section)

==Almost everywhere convergence==

In [[measure theory]], one talks about ''almost everywhere convergence''  of a sequence of [[measurable function]]s 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 [[Topological space|topology]] on the space of measurable functions on a [[measure space]] (although it is a [[Convergence space|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 (also known as the typewriter sequence), which are defined using the [[floor function]]: let <math>N = \operatorname{floor}\left(\log_2 n\right)</math> and <math>k = n</math> [[Modulo operation|mod]] <math>2^N,</math> and let
<math display=block>f_n(x) = \begin{cases}
1, & \frac{k}{2^N} \leq x \leq \frac{k+1}{2^N} \\
0, & \text{otherwise}.
\end{cases}</math>

Then any subsequence of the sequence <math>\left(f_n\right)_n</math> has a sub-subsequence which itself converges almost everywhere to zero, for example, the subsequence of functions which do not vanish at <math>x = 0.</math> But at no point does the original sequence converge pointwise to zero. Hence, unlike [[convergence in measure]] and [[Lp space|<math>L^p</math> convergence]], pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.