It is well know that if $f:\mathbb{R}^n\to\mathbb{C}$ is integrable, i.e. $\int_{\mathbb{R}^n}|f|dm<\infty$, then almost every point in $\mathbb{R}^n$ is a Lebesgue point of $f$, i.e. $$\lim_{r\to 0}\frac{1}{m(B(x,r))}\int_{B(x,r)}|f-f(x)|dm=0,$$ for almost all $x\in\mathbb{R}^n$.
What if $f$ is not integrable? Is there an example of a function $f$ that has no Lebesgue point on a set of positive measure? Can we find $f$ such that almost all $x$ is not a Lebesgue point? Or perhaps there is even a function that has no Lebesgue point at all!
