I'm currently solving exercise 10 of chapter 2 of Stein's Real Analysis. The problem is given as follows.
What I'm struggling to is the first part : $$\bigcup_{k = -\infty}^{\infty}F_k = \{x : f(x) > 0\}$$
The right direction is trivial because $F_k \subset \{x : f(x) > 0\}$ for all $k$. The thing is the left direction. Since $f$ has infinite values (on the set of $0$ measure), I think I have to consider the case $x$ with $f(x) = \infty$. But I can't assure that such $x$ is included in some of the $F_k$ because the value $f(x)$ of all elements $x$ in $F_k$ has finite values. The contraposition $\bigcap_k F_k^c \subset \{x : f(x) = 0 \}$ isn't clear as well.
So, any comments regarding the above statements would be appreciated. Thanks.