By definition, $G = \sigma Z + \mu$ for some $Z \in \text{Gaussian}(0, 1)$. Using result
R4593: Expectation of a standard gaussian random real number, we have
\begin{equation}
\mathbb{E} G
= \mathbb{E}(\sigma Z + \mu)
= \mu + \sigma \mathbb{E} Z
= \mu
\end{equation}
$\square$
