Casella and Berger's proof of Chebyshev's inequality relies on the statement that for random variable $X$, non-negative function $g(x)$, and $r > 0$,
$$ \int_{x: g(x)\geq r}^{\infty} g(x)f_X(x)dx \geq r\int_{x:g(x)\geq r}^{\infty}f_X(x)dx $$
I am struggling to understand the general validity of this. Can someone provide an explanation?
Perhaps it is more suggestive to rewrite the inequality in the form $$ \int_{x:g(x)-r\geq 0} (g(x)-r)f_X(x) \, dx \geq 0. $$ Then this says we want to show that if we integrate a function over the set of points where it is nonnegative, the result is nonnegative ($f_X(x) \geq 0$ since it is a the probability density, of course). This is one of those basic facts about integrals (usually going by the name of nonnegativity or monotonicity), and should be proved in any reasonable textbook.