# Fall 2016 (Neil-ish)

## 1

Define
$$
f(x) = \sum_{n=1}^{\infty} \frac{1}{n^{x}}.
$$ 

Show that $f$ converges to a differentiable function on $(1, \infty)$ and that
$$
f'(x)  =\sum_{n=1}^{\infty}\left(\frac{1}{n^{x}}\right)^{\prime}.
$$

> Hint:
$$
\left(\frac{1}{n^{x}}\right)^{\prime}=-\frac{1}{n^{x}} \ln n
$$

## 2

Let $f, g: [a, b] \to \RR$ be measurable with
$$
\int_{a}^{b} f(x) ~d x=\int_{a}^{b} g(x) ~d x.
$$

Show that either

1. $f(x) = g(x)$ almost everywhere, or
2. There exists a measurable set $E \subset [a, b]$ such that
$$
\int_{E} f(x) ~d x>\int_{E} g(x) ~d x
$$


## 3

Let $f\in L^1(\RR)$.
Show that
$$
\lim _{x \rightarrow 0} \int_{\mathbb{R}}|f(y-x)-f(y)| d y=0
$$

## 4
Let $(X, \mathcal M, \mu)$ be a measure space and suppose $\theset{E_n} \subset \mathcal M$ satisfies
$$
\lim _{n \rightarrow \infty} \mu\left(X \backslash E_{n}\right)=0.
$$

Define
$$
G \definedas \theset{x\in X \suchthat x\in E_n \text{ for only finitely many  } n}.
$$

Show that $G \in \mathcal M$ and $\mu(G) = 0$.


## 5

Let $\phi\in L^\infty(\RR)$. Show that the following limit exists and satisfies the equality
$$
\lim _{n \rightarrow \infty}\left(\int_{\mathbb{R}} \frac{|\phi(x)|^{n}}{1+x^{2}} d x\right)^{\frac{1}{n}} = \norm{\phi}_\infty.
$$

## 6

Let $f, g \in L^2(\RR)$. Show that
$$
\lim _{n \rightarrow \infty} \int_{\mathbb{R}} f(x) g(x+n) d x=0
$$