Let $\ \mu\ $be a nonzero finite Borel measure on $\ \mathbb{R}\ $,and let $\ F:\mathbb{R}\to\mathbb{R}\ $ be the function defined by $\ F(x)=\mu((-\infty,x]).\ $ Define a function $g$ on the interval $\ (0,\lim_{x\to +\infty}F(x))\ $ by $\ g(x)=inf\{t\in\mathbb{R}: F(t)\geq x\}. \ $
Question:Show that $\ g\ $ is Borel measurable.
I have already know that g is nondecreasing and finite valued.What I want to do is to find a representation of the inverse of $\ g\ $,or prove that $\ \{g\geq y\}\ $is Borel measurable,but I don’t know how to do that.
$g$ being nondecreasing is already almost the answer because for $x,y\in g^{-1}[r,s]$ with $x<y$, then for all $z$ with $x<z<y$, we have $r\leq g(x)\leq g(z)\leq g(x)\leq s$, so $z\in g^{-1}[r,s]$, so $g^{-1}[r,s]$ is an interval.