Open sets convergence
Consider the sequence of open sets of defined by
a) .
b) .
Study the convergence of the sets in the following types of convergence:
 weak convergence of characteristic functions;
 convergence in the topology defined by the Hausdorff distance;
 convergence in the way of compact sets.
(Each of the previous types of convergence will be presented below.)
weak convergence of characteristic functions
The sequence of measurable sets of is said to converge in the way of characteristic functions towards the measurable set if the sequence of characteristic functions converges to in .
The first remark is that doesn’t matter since . The second remark is that if the sequence converges in the weak* topology of to then converges in the way of characteristic functions to .
convergence in the topology defined by the Hausdorff distance
Consider a fixed (large enough) compact of . For two compacts in , and consider the following definitions:
 ;
 ;
 .
A sequence of open sets and included in . We say that converges in the Hausdorff distance to if for . We denote . We can define the same type of definition for compact sets from .
convergence in the way of compact sets
We say that the sequence of open sets of converges in the way of compact sets towards , and we denote if we have
 compact , we have for large enough;
 compact we have for large enough.
Solution: a) For the characteristic functions convergence we will show that weakly in , which gives us an example of sequence of characteristic functions whose limit is not a characteristic function.
We need to show that , for all .
For proving this, we use a trick used in many real analysis proofs. Start with step functions, then extend to positive function, and finally to measurable functions. I will propose a similar, more detailed approach:
 where are fractions with denominators powers of . This is quite easy, given the form of the sets .
 extend the last step to all compact intervals, $latex \phi=\chi_{[a,b]}$, (using Beppo Levi’s Theorem)
 extend to step functions
 then to positive functions
 finally to integrable functions
b) , therefore, the convergence in the way of characteristic functions is clear enough.
Denote which converges in the Hausdorff distance to the interval , which means by taking complements that .
Finally, it is clear that does not converge to any open set in the way of compacts. Suppose . If . Pick . Then it would mean that for large enough. But the “holes” in become “denser” in , which means that if we pick some compact containing an open set, that compact will not satisfy the given definition. On the contrary, if the sequence would converge to a nonempty open set, some compact within it will not belong to all .

November 25, 2010 at 5:38 pmDomains convergence « Problems – Beni Bogoşel