theorem 10.14 Browder Mathematical Analysis

Question

From Mathematical Analysis Andrew Browder

If $g$,$h$ not comparable,and $\nu_g$,$\nu_h\in \mathscr{M}$ how can we find $\nu_f \in \mathscr{M}$ such that $\nu_g \leq \nu_f$ and $\nu_h \leq \nu_f$.That is,If $g$,$h$ not comparable,how collection $\mathscr{M}$ will satisfies that hypothesis of $9:13$

Thanks in Advance!!

Answer 1

The author notes that if $g$ and $h$ are simple functions less than $f$, so is $\max(g,h)$.

Then $\nu_{\max(g,h)}$ plays the role of $\mu_3$ for theorem 9.13

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I hope this helps ^_^