Let $f(x)$ be a non-negative measurable function. Assume that the function $xf^2(x)$ is integrable on the whole real line. Prove that the function $x$ is integrable on the set $M_y = \{x:f(x)>y\}$ for every $y>0$.
I am not exactly sure where to begin. I was thinking about trying to show that $\mu (M_y)$ was finite then trying to approximate using intervals but wasn't sure how I could conclude that. I am sure I am missing something simple. I don't really want an answer but more of a hint to push me in the right direction.
It's pretty hard to give a hint which will not get you to the answer. Well, by the definition of Lebesgue integration we know that the function $xf^2(x)$ is integrable on both intervals $[0,\infty)$ and $(-\infty,0)$. Alright, so we have:
$$\int_{[0,\infty)}xf^2(x)d\mu\geq \int_{M_y\cap[0,\infty)} xf^2(x)d\mu\geq y^2\int_{M_y\cap [0,\infty)}xd\mu$$
Now do the same for the function $-xf^2(x)$ on $(-\infty,0)$. Can you finish from here?