It's a bulky four part question, most of which I've already worked through, but towards the end it begs for the construction of sets based on triangle functions and rectangular functions, which has so far strained my abilities. Here is the question:
Let $\{r_n\}$ be a dense subset of $\Bbb R$. Let $g_n:\Bbb R \rightarrow [0,1]$ be continuous $\forall n\in\Bbb N$. Assume $g_n(r_n)=1$ $\forall n$. Assume $\sum_{n\in\Bbb N} m(\{g_n>0\})<\infty$. Let $g=\sum_{n\in \Bbb N}g_n$.
- Show that $g$ is defined and finite almost everywhere.
- Show that $g$ is not bounded on any open interval (non-trivial)
- Let $g_n$ be a triangle function with height 1 and base $\frac 1 {2^n}$. Explicitly construct a set $F \subset [0,1]$ such that $m([0,1]\setminus F)<\frac 1 8$ and that $g_n \rightarrow 0$ on $F$ almost everywhere.
- Let $g_n$ be a rectangular function with height 1 and base $\frac 1 {2^n}$. Explicitly construct a set $F \subset \Bbb R$ such that $m(\Bbb R \setminus F)<\frac 1 8$ and a continuous function $f$ on $\Bbb R$ where $g=f$ on $F$.
So yes it's a long one. My work so far towards 1 and 2 are posted below. Being checked over would be great!
3 and 4 have stumped me pretty good, I am having trouble relating a triangle/rectangle function to the given conditions.
My Solutions in an answer