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.

Measure Theory – Problems

Measure Theory – Solutions

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

Erratum:

In problem 8, there should be a in the definition of the set .

Analysis Prelim Notes

Real Analysis Problems

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