. (a) Let $f : \mathbb R \to [0,\infty)$ be measurable, and assume that $$\int f\,dm < \infty,$$ where $m$ is the Lebesgue measure. Prove that there is a nonincreasing function $g : [0,\infty) \to [0,\infty)$ such that $$ m(\{x : f(x) < a\}) = m(\{x : g(x) < a\})\text{ for all }a > 0. $$ (b) Prove that if $f, g : [0, 1] → [0,\infty)$ are two measurable functions such that (a) holds, then $$\int_{[0,1]} f\, dm = \int_{[0,1]} g\,dm$$.
Thank you so much. I just have no idea how to start.
