Homework 4, Real Analysis 2

Due March 2

Problem 1

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

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

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

Problem 2

1. Let $F\subset\R$ be a closed set such that $m(\R\setminus F) <\infty$, and let $\delta(x) = d(x,F)$ be the distance from x to F. Then $\delta(x)$ is a Lipschitz function.
2. Let $\displaystyle I(x) = \int_\R \frac{\delta(x)}{|x-y|^2} dy.$ Then $I(x) = \infty$ for any $x\in\R\setminus F$, and $I(x) < \infty$ for a.e $x\in F$.

Problem 3

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

Problem 4

Consider the function defined on $\R$ by

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

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

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

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.
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