The function defined with the Lebesgue's measure is measurable

Question

I have to show that for $f: \mathbb{R} \rightarrow\ [0,\infty)$ a measurable function, the function $g: \mathbb{R} \rightarrow \ [0,\infty], \ g(r):=\lambda(\{f>r\})$ is measurable. I noted that the function $g$ is strictly decreasing, but I don't know how to continue, any suggestions? Thanks!!

Answer 1

As you have noted, $g$ is strictly decreasing. But this already implies measurability, since the set $Q_{r_0} := \{r : g(r) > r_0\}$ must be an interval for all $r_0$ (show this!), and every interval is $\lambda$-measurable.