# Spring 2018

## 1
Define
$$
E:=\left\{x \in \mathbb{R}:\left|x-\frac{p}{q}\right|<q^{-3} \text { for infinitely many } p, q \in \mathbb{N}\right\}.
$$

Prove that $m(E) = 0$.

## 2
Let
$$
f_{n}(x):=\frac{x}{1+x^{n}}, \quad x \geq 0.
$$

a. Show that this sequence converges pointwise and find its limit. Is the convergence uniform on $[0, \infty)$?

b. Compute 
$$
\lim _{n \rightarrow \infty} \int_{0}^{\infty} f_{n}(x) d x
$$

## 3
Let $f$ be a non-negative measurable function on $[0, 1]$. 

Show that
$$
\lim _{p \rightarrow \infty}\left(\int_{[0,1]} f(x)^{p} d x\right)^{\frac{1}{p}}=\|f\|_{\infty}.
$$

## 4
Let $f\in L^2([0, 1])$ and suppose
$$
\int_{[0,1]} f(x) x^{n} d x=0 \text { for all integers } n \geq 0.
$$
Show that $f = 0$ almost everywhere.


## 5

Suppose that

- $f_n, f \in L^1$,
- $f_n \to f$ almost everywhere, and
- $\int\left|f_{n}\right| \rightarrow \int|f|$.

Show that $\int f_{n} \rightarrow \int f$