Real analysis Qualifying exam, January 2021

DO NOT WRITE YOUR NAME ON YOUR WORK

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In order to receive the full credit for a problem, a detailed argument (rather than a sketch of the proof) is needed. Whenever applying one of the standard theorems, please indicate that clearly. Full solutions on a smaller number of problems will be worth more than partial solutions on more problems.

Problem 1

Let $f_{n}, n \geq 1$, and $f$ be measurable functions on a space ( $\Omega, \mathcal{F}, \mu$ ), such that $f_{n} \rightarrow f$ in measure. Does this imply that there exists a measurable set $A \subseteq \Omega$ with $\mu(\Omega \backslash A)=0$ such that $f_{n}(x) \rightarrow f(x)$ for all $x \in A$ ?
If yes, prove this. If no, give a counterexample.

Problem 2

Let $B$ be a measurable subset of the two-dimensional plane such that the intersection of $B$ with every vertical line is finite or countable. Find $\mu(B)$, where $\mu$ is the two-dimensional Lebesgue measure. Justify your answer.

Problem 3

Let ( $\Omega, \mathcal{F}$ ) be a measurable space, and $\mu, \nu, \rho$ be three finite positive measures on ( $\Omega, \mathcal{F}$ ) such that $\mu \ll \nu$ (i.e., $\mu$ is absolutely continuous with respect to $\nu$ ). Show that there exists a measurable function $f$ on $\Omega$ such that for all $E \in \mathcal{F}$ we have

$$\mu(E)=\int_{E} f d \nu+\int_{E}(f-1) d \rho$$

(Hint: use Radon-Nikodym's Theorem)

Problem 4

Let $f, g$ be nonnegative measurable functions on $[0,1]$, and $a, b, c, d \geq 0$ be arbitrary nonnegative numbers. Show that then
$\left(a c+b d+\int_{0}^{1} f(x) g(x) d x\right)^{3} \leq\left(a^{3}+b^{3}+\int_{0}^{1}(f(x))^{3} d x\right)\left(c^{3 / 2}+d^{3 / 2}+\int_{0}^{1}(g(x))^{3 / 2} d x\right)^{2}$.
Partial credit is given for proving the inequality in the particular case $a=b=c=d=0$.

Problem 5

Let $f(x)$ be a continuous function on $[0,1]$. Show that for every $\varepsilon>0$ there exists $n \in \mathbb{Z}_{\geq 0}$ and constants $a_{0}, a_{1}, \ldots, a_{n} \in \mathbb{R}$ such that for the differential operator

$$D:=\sum_{k=0}^{n} a_{k}\left(\frac{d}{d x}\right)^{k}=a_{0}+a_{1} \frac{d}{d x}+a_{2}\left(\frac{d}{d x}\right)^{2}+\ldots+a_{n}\left(\frac{d}{d x}\right)^{n}$$

we have $\left|f(x)-e^{x^{2}}\left(D e^{-x^{2}}\right)\right|<\varepsilon$ for all $x \in[0,1]$. Here $e^{x^{2}}\left(D e^{-x^{2}}\right)$ is the function obtained by applying $D$ to $e^{-x^{2}}$ and after that multiplying the result by $e^{x^{2}}$.