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.
Integration and Limits – Problems
Integration and Limits – Solutions
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. .
Real Analysis Problems – Convergence in Measure and Fubini
Hints – Convergence in Measure and Fubini
In problem 8, there should be a in the definition of the set .