Function 1/x is not locally integrable

Question

I have to show, that $\frac{1}{x}$ is not locally integrable on $\mathbb{R}$

How to show it formally?

Answer 1

Recall that a function $f$ is locally integrable on $\mathbb R$ if for each compact subset $K$ of $\mathbb R$, $f\chi_K$ is integrable.

Here, taking $K=[0,1]$ we can see that $f$ is not integrable on $K$ (the value at $0$ does not matter as we work almost everywhere). Indeed, we can either compute $\int_{\varepsilon}^1\frac 1x\mathrm dx$ or pick a sequence of simple functions smaller than $f$ and such that $\int_K f_n(x)\mathrm dx\geqslant n$.