Let $h$ be any other member of $C$. To show that $g$ is a minimizer, you need to establish that $I(g) \leq I(h)$, which is equivalent to showing
$$
\int f(g-h) \leq 0.
$$
To show this, split the range of integration into the sub-level, sup-level, and level sets of $f$; i.e., $\{fs\}$, and $\{f=s\}$:
\begin{align}
\int f (g-h)
&= \int\limits_{\{fs\}} f (g - h) + \int\limits_{\{f=s\}} f (g - h) \\
&\leq s \int\limits_{\{f s\}} f h + \int\limits_{\{f=s\}} s (g-h) \\
&\leq s \int\limits_{\{f s\}} h + s \int\limits_{\{f=s\}} (g-h) \\
& \leq s \left( \int\limits_{\{fs\}} (g-h) + \int\limits_{\{f=s\}} (g-h) \right) \\
& = s \int(g - h) = s (G -G) = 0.
\end{align}