Since $g$ is unsigned, result
R3531: Pointwise product with indicator function is lower bound for unsigned basic function shows that $g I_{\{ f < g \}} \leq g$. Using
R3529: Tautological pointwise inequality from indicator functions on sublevel sets, we then conclude
\begin{equation}
f I_{\{ f < g \}} \leq g I_{\{ f < g \}} \leq g
\end{equation}
$\square$
