I'm working on a problem in measure theory:
Find the Lebesgue set $L_f=\{x:\lim_{r\to 0} \frac{1}{m(B(r,x))} \int_{B(r,x)}\lvert f(y)-f(x)\rvert dy=0\}$ if:
- $f:\mathbb R^n \to \mathbb C$ is continuous
- $f=\chi_{E}$, where $E\in\mathbb R^n$ is Lebesgue null; $m(E)=0$. $\chi$ denotes the indicator function.
- $f(x)=\lfloor x\rfloor$ on $\mathbb R$
I'm having a bit of a tough time wrapping my head around what the Lebesgue sets really are, and consequently is struggling to characterize the Lebesgue sets w.r.t to these functions. I could really use some help on these!
If $f$ is continuous at $x$ then $|f(y)-f(x)| <\epsilon$ foe all $y \in B(x,r)$ provided $r$ is small enough. Hence $x$ is a Lebesgue point. In 2) $f=0$ a.e. so you can simply repalce $f(y)$ by $0$ in the definition without changing the integral. Hence $x$ is a Lebesgue point iff $f(x)=0$ which means $x \in E^{c}$. (There was mistake in my comment. I meant $x \in E^{c}$ not $x \in E$). In 3) all points except integers are points of continuity and hence they are Lebesgue points. At integer points it is easy to see that the limit in the definition equals $\frac 1 2$, not $0$.