# Fall 2018

## 1
Let $f(x) = \frac 1 x$.
Show that $f$ is uniformly continuous on $(1, \infty)$ but not on $(0,\infty)$.

## 2
Let $E\subset \RR$ be a Lebesgue measurable set.
Show that there is a Borel set $B \subset E$ such that $m(E\setminus B) = 0$.

## 3
Suppose $f(x)$ and $xf(x)$ are integrable on $\RR$.
Define $F$ by
$$
F(t):=\int_{-\infty}^{\infty} f(x) \cos (x t) d x
$$
Show that 
$$
F^{\prime}(t)=-\int_{-\infty}^{\infty} x f(x) \sin (x t) d x.
$$


## 4
Let $f\in L^1([0, 1])$.
Prove that
$$
\lim_{n \to \infty} \int_{0}^{1} f(x) \abs{\sin n x} ~d x= \frac{2}{\pi} \int_{0}^{1} f(x) ~d x
$$

> Hint: Begin with the case that $f$ is the characteristic function of an interval.

## 5
Let $f \geq 0$ be a measurable function on $\RR$.
Show that
$$
\int_{\mathbb{R}} f=\int_{0}^{\infty} m(\{x: f(x)>t\}) d t
$$

## 6
Compute the following limit and justify your calculations:
$$
\lim_{n \rightarrow \infty} \int_{1}^{n} \frac{d x}{\left(1+\frac{x}{n}\right)^{n} \sqrt[n]{x}}
$$