# A function finite almost everywhere but NOT Lebesgue integrable

I was trying to solve a problem on measure theory from the book of Folland and got stuck in one problem(Prob-25,page-59). My comments are in italics.

Let $$m$$ be the Lebesgue measure on $$\mathbb{R}$$ and $$L^1(m)$$ be the space of all m'ble functions $$g$$ such that $$|g|$$ is integrable.

Problem: Let $$f(x)=x^{-\frac{1}{2}}$$ for $$0 and $$f(x)=0$$ else. Let $$\{r_n\}_{n=1}^\infty$$ be an enumeration of rationals.

Define $$g(x)=\sum_n\frac{f(x-r_n)}{2^n}$$.

Show that

(1) $$g\in L^1(m)$$.

(2) $$g$$ is discontinuous everywhere and unbounded on every interval. It remains so after a correction on a set of measure $$0$$.

(3) $$g^2<\infty$$ a.e. but $$g^2$$ is not integrable over any interval.

Since $$f$$ is non negative, by applying monotone convergence theorem I could solve 1. But I'm completely blind about 2 and 3 even about how to start. Any kind of suggestions will be appreciated.

Thank you for your help in advance.

## 1 Answer

Hint: On the interval $$(a,b)$$, pick a rational $$r$$ such that $$a. Note that $$g(x) \ge 2^{-n} f(x-r)$$, where $$n$$ is such that $$r = r_n$$.

• Since $f\geq 0$, indeed $g(x)\geq 2^{-n}f(x-r_n)$ for all $n$.Did you mean this inequality for $g$ to be discontinuous near any $r_n \in (a,b)?$. Then it solves the $2$nd one.Thank you.. Commented Jul 21, 2020 at 6:12
• It solves the third one as well. Think about $\int_{(a,b)} f(x-r_n)^2 \, dx$. Commented Jul 21, 2020 at 6:13
• But what's about the 3rd? I can't see the solution coming out of your argument? Would you please explain? Commented Jul 21, 2020 at 6:14