Problem set 8, Harmonic Analysis

Due March 29

Problem 1

$A\subset\mathbb R^d$ is measurable if and only if, for all $B\subset\mathbb R^d$, 

$|B|_* = |B\cap A|_* + |B\setminus A|_*$.

Problem 2

Let $A\subset\mathbb R^d$. The following are equivalent.

1. $A$ is measurable.
2. $A = P\setminus M$, where $P$ is a $G_\delta$ set and $|M|=0$.
3. $A = Q\cup N$, where $Q$ is an $F_\sigma$ set and $|N|=0$.

Problem 3

Let $A\subset\mathbb R^d$ be a measurable set.

1. For $\delta > 0$, let $\delta A = \{ \delta x: x\in A\}$. Then $\delta A$ ls measurable and $|\delta A| = \delta^d |A|$.
2. For a $d$-tuple $\bar{\delta} = (\delta_1, \ldots, \delta_d)$ with each $\delta_j>0$, $j=1,\ldots,d$, define $\bar\delta A = \{(\delta_1 x_1, \ldots, \delta_d x_d): (x_1, \ldots, x_d)\in A\}$. Then $\bar\delta A$ is measurable and $|\bar\delta A| = \delta_1\cdots\delta_d |A|.$

Problem 4

If $f:\mathbb R\to\mathbb R$ is monotone, then it is measurable.

Problem 5

If $f_n\ge 0$ are measurable on $\mathbb R^d$, $f_n\to f$ pointwise and $\int f = \lim\int f_n < \infty$, then

$\displaystyle \int_A f_n \to \int_A f$

for all measurable $A\subset\mathbb R^d$.

This might not be true if $\int f = \lim\int f_n = \infty$.