Q:

About positive functions on an infinite set

Let $F\in BV([0,1])$, $F\geq 0$, and $F\gt 0$ on an infinite set.
Can we have $F=0$ almost everywhere w.r.t. Lebesgue measure?
I'd like to say no but I can't prove it...

A:

I think, we can. Take a kind of Dirichlet-type function
$$f(x)=\begin{cases}
1&\hbox{ for } x\in Q \cap [0,1]\\
0&\hbox{ for } x\in (R\setminus Q)\cap [0,1]
\end{cases}
$$
This function is greater than zero on a countable set, but it is equal zero almost everywhere.

