I'm reading a few papers and trying to understand why one paper is supposed to be a strengthening of the other. The bulk of the papers didn't have to do with what follows, but at one point they imply that an a.e. essentially continuous characteristic function is automatically a characteristic function of a set which is a.e. an open set, and this is stronger. I'll make the definitions rigorous:
We say $x \in \mathbb{R}^n$ is a point of $\textit{essential continuity}$ for $f$ (a function on $\mathbb{R}^n$) if for every $\varepsilon > 0$, there exists a neighborhood $U_{x,\varepsilon}$ of $x$ such that $|f(y)-f(x)| < \varepsilon$ for a.e. $y \in U_{x,\varepsilon}$ (hence the "essentially" continuous). I have two main questions (and one side question which is less important):
Suppose the characteristic function $1_E$ is a.e. essentially continuous, that is, for a.e. $x$, $x$ is a point of essential continuity for $1_E$. Then why must $E$ be a.e. equivalent to an open set?
For this question, I was thinking of letting $\varepsilon < 1$ be fixed, and then the definition of essential continuity implies that for a.e. $x$, there is a neighborhood $U_x$ of $x$ such that for a.e. $y \in U$, we have $|1_E(x)-1_E(x)| < 1$. In particular, this means either both $x$ and $y$ are in $E$, or both are not in $E$. I tried to let my open set $U$ be the union of all the $U_x$ for $x \in E$ which are points of essential continuity, but the problem is there may be points $y$ in each $U_x$ which are not in $E$, and there could be uncountably many $U_x$. Thus the resulting $U$ is not in fact open.
Is there a good example of a set $E$ which is a.e. equivalent to an open set, but $1_E$ is not a.e. essentially continuous?
I don't really know where to start with this one, but I imagine we require a highly disjoint set, like some sort of fat cantor set. This particular set won't work though, since the fat Cantor set, say $K$, is not equivalent to an open set, but perhaps the complement $[0,1] \setminus K$ will work - I'm stuck on this too though.
I'm curious if we can strengthen 1 and 2: with the hypothesis of 1, why must both $E$ and $E^c$ be a.e. equivalent to open sets? And for 2, does there exist a set $E$ a.e. equivalent to an open set such that $E^c$ is also a.e. equivalent to an open set, and $1_E$ is not a.e. essentially continuous?
The motivation for 3 is that I believe the first paper's result can be strengthened from a statement about $E$ being a.e. equivalent to an open set, to a statement about both $E$ and $E^c$ being equivalent to an open set. If so, then in order for the second paper to be a true generalization, point 3 must hold.
Any help is appreciated, thanks.