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Homework 4, Real Analysis 2

Due March 2

Problem 1

Let f be integrable on $latex [0,b]$ and define, on $latex [0,b]$,

$latex \displaystyle g(x) = \int_x^b \frac{f(t)}{t} dt.$

Then g is integrable on $latex [a,b]$ and $latex \displaystyle \int_0^b g(x) dx = \int_0^b f(t) dt$.

Problem 2

  1. Let $latex F\subset\R$ be a closed set such that $latex m(\R\setminus F) <\infty$, and let $latex \delta(x) = d(x,F)$ be the distance from x to F. Then $latex \delta(x)$ is a Lipschitz function.

  2. Let $latex \displaystyle I(x) = \int_\R \frac{\delta(x)}{|x-y|^2} dy.$ Then $latex I(x) = \infty$ for any $latex x\in\R\setminus F$, and $latex I(x) < \infty$ for a.e $latex x\in F$.

Problem 3

There exists $latex f\in L^1(\R^d)$ and a sequence $latex f_n\in L^1(\R^d)$ such that $latex f_n\to f$ in $latex L^1$, but $latex f_n(x) \not\to f(x)$ for every x.

Problem 4

Consider the function defined on $latex \R$ by

$latex f(x) = \begin{cases} x^{-1/2} & 0 < x < 1,\\0 & \text{otherwise.} \end{cases}$

For a fixed enumaeration $latex \{r_n\} = \Q$, let

$latex \displaystyle F(x) = \sum_{n=1}^\infty 2^{-n} f(x - r_n)$.

  1. F is integrable, and thus the series for F converges for a.e. x.

  2. F is unbounded on every interval.

  3. If g = F a.e., then g is unbounded in any interval.