real analysis - Increasing Sequence of Step Functions Converging to Characteristic Function of Fat Cantor Set

This is related to this question.

I want to find an open set $G\subseteq [a,b]$ such that there does not exist an increasing sequence of step functions on $[a,b]$ which converges almost everywhere to $\chi_{G^c}$.

By that question, if $G^c$ has interior, I can find an increasing sequence $s_n$ of step functions which converges to the characteristic function of the interior. If the noninterior points form a set of measure zero, the sequence will still converge almost everywhere.

So there are two options. A nowhere dense closed set which has positive measure, or a closed set with nonempty interior but which boundary has not zero measure.

For the first case, I found this, a Fat Cantor Set, and from a little answer here for this case I should be able to get the example.

So we can propose this

Proposition. Let $F\subseteq [0,1]$ be the fat Cantor Set. Then there does not exist an increasing sequence $s_n$ of step functions such that $s_n\nearrow \chi_F$ almost everywhere in $[0,1]$.

Is there any idea of how can I prove that inexistence?

The second case seems to be around something similar.