(Expanding the previous post to include another question): I've spent quite some time but couldn't understand the following 3 highlighted expressions (as per screenshot). @Azif00 kindly explained to me the first two (inequalities). Could someone please help with the third one (equality)? Thanks
$g$ will decrease as $x$ increases in value. Therefore, as $g(x)$ is a decreasing function, then $g^{-1}(g(x)) = x$ is increasing.
So if set for probability changes from $g(x) \leqslant y$ then $x \geqslant y$. So the new limits for integral change from "infinity to $x$" to "$x$ to infinity".
C.D.F. of $y$
= C.D.F. of $x$ at infinity - C.D.F. of $x$ at $x$
$= 1 -Fx(X = x).$