Real Analysis Prelim Camp

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 \epsilon > 0 and n \in \mathbb{N} define the sets A_n^\epsilon := \{x\in[0,1] : |f_n(x)| \geq \epsilon\}. Then Chebyshev implies that 

|A_n^\epsilon| \leq \frac{1}{\epsilon} \Vert f_n \Vert_{L^1([0,1])} \leq \epsilon^{-1} n^{-2}

Then Borel-Cantelli implies that \limsup_n A_n^\epsilon has measure zero. Now, define the set A : = \bigcup_{k =1}^\infty A_n^{1/k}. Note that A has measure zero and that x \in A if and only if f_n(x) doesn’t converge to zero. This shows that f_n(x) \rightarrow 0 for a.e. x \in[0,1].


Real Analysis Problems – Convergence in Measure and Fubini

Hints – Convergence in Measure and Fubini

Erratum:

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


Analysis Prelim Notes

Real Analysis Problems

 

 

 

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