Here is where I will upload notes, problems, and solutions for the real analysis prelim camp in Summer, 2020. All the problems come from past UT prelim exams.
For problem 11, there is actually an alternative way using Borel-Cantelli. For each and define the sets . Then Chebyshev implies that
Then Borel-Cantelli implies that has measure zero. Now, define the set . Note that has measure zero and that if and only if doesn’t converge to zero. This shows that for a.e. .
In problem 8, there should be a in the definition of the set .