Suppose a property holds in every compact subset of $I$. Does it follow that this property holds almost everywhere on $I$?
I want to use the dominated convergence theorem but do not know almost anything about Lebesgue Measure. I know that the function series converges on every compact subset of $(0,1)$, does it follow that this series converges almost everywhere on $(0,1)$?
Actually it is my fault to ask this question not in more proper way, only if I know much about Lebesgue Measure.
Suppose there is a set of positive measure in which you have not convergence. Then it exists a compact set $[ \epsilon, 1-\epsilon]$ such that its intersection with the set considered before has positive measure. But the function converges on $[ \epsilon, 1-\epsilon]$, and you have a contradiction. This should actually prove that the function converges everywhere, and it is based on the fact that you can view your open interval as a countable increasing union of compact sets