Real analysis Qualifying exam, August 2020

Make sure that you have signed the Honor Pledge on Collab.
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.

Problem 1

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous, almost everywhere differentiable, and such that $f^{\prime}(x)=1$ almost everywhere. (Both "almost everywhere" properties are assumed with respect to the Lebesgue measure on $\mathbb{R}$.) Does this imply that $f(2)-f(1)=1$ ?

If yes, prove this. If no, give a counterexample.

Problem 2

Is every open set in $\mathbb{R}^{2}$ a countable union of closed sets?

If yes, prove this. If no, give a counterexample.

Problem 3

Let $\mathcal{H}$ be a separable complex Hilbert space with basis (complete orthonormal system) $f_{1}, f_{2}, f_{3}, \ldots$ Define a linear operator $P$ in $\mathcal{H}$ by setting

$$P\left(f_{n}\right)=f_{n+1}, \quad n=1,2, \ldots$$

(a) Find the adjoint $P^{*}$ to $P$.
(b) Find the operators $P P^{*}$ and $P^{*} P$.

Problem 4

Let $(X, \mathcal{F}, \mu)$ be a measure space with $\mu(X)=1$. Let $f_{n}: X \rightarrow \mathbb{R}$ be measurable functions such that for all $t \in \mathbb{R}$,

$$\lim _{n \rightarrow+\infty} \mu\left(x: f_{n}(x) \leq t\right)= \begin{cases}0, & t<0 \\ 1, & t \geq 0\end{cases}$$

Show that $f_{n} \rightarrow 0$ in measure.

Problem 5

Show that the operator

$$(T f)(x):=\int_{0}^{\infty} \frac{f(y)}{x+y} d y$$

is bounded in the space $L^{p}\left(\mathbb{R}_{\geq 0}\right)$ for all $1<p<+\infty$.