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.
Post Tagged with 

Comments & Responses

Leave a Reply