# 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