The existence of a set with nonzero measure on which two measurable functions $f>g$ is "separated" by a constant.

Question

Suppose $(\Omega,\mathscr{F},\mu)$ is a measure space and $f,g$ are $(\mathscr{F},\mathscr{B})-$measurable functions where $\mathscr{B}$ is the Borel algebra on $\mathbb{R}$. If $$ \mu(\{f<g\})>0 $$ then are we able to find a constant $\xi\in\mathbb{R}$ such that the set $\{f\leq \xi< g\}$ is also of nonzero measure? I am confident that I could find $\epsilon>0$ such that the set $\{f<g-\epsilon\}$ is of nonzero measure, but I find the existence of such a constant $\xi$ questionable.

Answer 1

The rational numbers can be denoted as $r_1,r_2,...$

Denote $A_k=\{f\le r_k <g\}.$

Then $\cup_kA_k=\{f < g\}$.

So you can find one $A_k$ with positive measure.