By definition
\begin{equation}
\int_X f \, d \delta_{x_0}
= \int_X f^+ \, d \delta_{x_0} - \int_X f^- \, d \delta_{x_0}
\end{equation}
We may thus use results
to conclude
\begin{equation}
\int_X f \, d \delta_{x_0}
= \int_X f^+ \, d \delta_{x_0} - \int_X f^- \, d \delta_{x_0}
= f^+(x_0) - f^-(x_0)
= f(x_0)
\end{equation}
$\square$
