Lemma 5.24.405BCBF7FAC9C

Proof. Choose a number $c$ with $a(x)<c<b(x)$. By upper semicontinuity, $a<c$ on a neighborhood $V$ of $x$. If $\{ b>c\} \cap V$ had zero volume, then $b\leq c$ almost everywhere on $V$, and lemma 5.24.3 would give $b(x)\leq c$, a contradiction. Thus $\{ b>c\} \cap V$, which is contained in $\{ a<b\} $, has positive volume. $\square$