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Problem set 8, Harmonic Analysis

Due March 29

Problem 1

$latex A\subset\mathbb R^d$ is measurable if and only if, for all $latex B\subset\mathbb R^d$, 
$latex |B|_* = |B\cap A|_* + |B\setminus A|_*$.

Problem 2

Let $latex A\subset\mathbb R^d$. The following are equivalent.
  1. $latex A$ is measurable.
  2. $latex A = P\setminus M$, where $latex P$ is a $latex G_\delta$ set and $latex |M|=0$.
  3. $latex A = Q\cup N$, where $latex Q$ is an $latex F_\sigma$ set and $latex |N|=0$.

Problem 3

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

Problem 4

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

Problem 5

If $latex f_n\ge 0$ are measurable on $latex \mathbb R^d$, $latex f_n\to f$ pointwise and $latex \int f = \lim\int f_n < \infty$, then
$latex \displaystyle \int_A f_n \to \int_A f$
for all measurable $latex A\subset\mathbb R^d$.
This might not be true if $latex \int f = \lim\int f_n = \infty$.