# Spring 2014

## 1

1. Give an example of a continuous $f\in L^1(\RR)$ such that $f(x) \not\to 0$ as$\abs x \to \infty$.

2. Show that if $f$ is *uniformly* continuous, then
$$
\lim_{\abs x \to \infty} f(x) = 0.
$$

## 2
Let $\theset{a_n}$ be a sequence of real numbers such that
$$
\theset{b_n} \in \ell^2(\NN) \implies \sum a_n b_n < \infty.
$$
Show that $\sum a_n^2 < \infty$.

> Note: Assume $a_n, b_n$ are all non-negative.


## 3
Let $f: \RR \to \RR$ and suppose
$$
\forall x\in \RR,\quad f(x) \geq \limsup _{y \rightarrow x} f(y)
$$
Prove that $f$ is Borel measurable.


## 4
Let $(X, \mathcal M, \mu)$ be a measure space and suppose $f$ is a measurable function on $X$.
Show that
$$
\lim _{n \rightarrow \infty} \int_{X} f^{n} ~d \mu =
\begin{cases}
\infty & \text{or} \\
\mu(f\inv(1)),
\end{cases}
$$
and characterize the collection of functions of each type.

## 5
Let $f, g \in L^1([0, 1])$ and for all $x\in [0, 1]$ define
$$
F(x):=\int_{0}^{x} f(y) d y \quad \text { and } \quad G(x):=\int_{0}^{x} g(y) d y.
$$

Prove that
$$
\int_{0}^{1} F(x) g(x) d x=F(1) G(1)-\int_{0}^{1} f(x) G(x) d x
$$