If $X$ is an empty set, then the inequality is vacuously true everywhere, so assume that $X$ is nonempty and let $x \in X$.
If $f(x) < g(x)$, then $I_{\{ f < g \}}(x) = 1$ and
\begin{equation}
f(x) I_{\{ f < g \}}(x) = f(x) < g(x) = g(x) I_{\{ f < g \}}(x)
\end{equation}
If $f(x) \geq g(x)$, then $I_{\{ f < g \}}(x) = 0$ and
\begin{equation}
f(x) I_{\{ f < g \}}(x) = f(x) \cdot 0 = 0 = g(x) \cdot 0 = g(x) I_{\{ f < g \}}(x)
\end{equation}
$\square$
