Since $x \not\in \emptyset$ for all $x \in X$, then we have $I_{\emptyset}(x) f(x) = 0$ for all $x \in X$. Hence, applying result
R3515: Unsigned basic integral zero iff function almost everywhere zero, we have
\begin{equation}
\int_{\emptyset} f \, d \mu
= \int_X I_{\emptyset} f \, d \mu
= 0
\end{equation}
$\square$
