$\DeclareMathOperator\intr{int}\DeclareMathOperator\meas{meas}$This is one—probably not the best—way of thinking about it.  (I am used to the $p$-adic world, where one can think about (real-valued) measures much less subtly than in the real world.)  I hope someone will come along and write something more elegant.

The way we know we have the correct measure on $G(F)\backslash G(\mathbb A)$ is that $\int_{G(F)\backslash G(\mathbb A)} \int_{G(F)} f(h g)\mathrm dh\,\mathrm d\dot g = \int_{G(\mathbb A)} f(g)\mathrm dg$ for all $f \in \operatorname C_c^\infty(G(\mathbb A))$.  Now take $\mathfrak S$ and $\mathfrak S'$ such that $G(\mathbb A) = G(F)\mathfrak S'$ and $\mathfrak S' \subseteq \intr(\mathfrak S)$.  Write $\mathfrak S'$ and $\mathfrak S$ as countable, increasing unions of compacts $\mathfrak S'_k$ and $\mathfrak S_k$ such that $\mathfrak S'_k \subseteq \intr(\mathfrak S_k)$ for all $k$; and, for each $k$, let $f_k \in \operatorname C_c^\infty(G(\mathbb A))$ be a non-negative function that is $1$ on $\mathfrak S'_k$ and $0$ outside $\mathfrak S_k$.  Then
\begin{multline*}
\meas_{\mathrm d\dot g}(G(F)\backslash G(\mathbb A))
= \lim_{k \to \infty} \meas_{\mathrm d\dot g}(G(F)\backslash G(F)\mathfrak S'_k)
\le \lim_{k \to \infty} \int_{G(F)\backslash G(\mathbb A)} \int_{G(F)} f_k(h g)\mathrm dh\,\mathrm d\dot g \\
= \lim_{k \to \infty} \int_{G(\mathbb A)} f_k(g)\mathrm dg
\le \lim_{k \to \infty} \meas_{\mathrm dg}(\mathfrak S_k)
= \meas_{\mathrm dg}(\mathfrak S) < \infty.
\end{multline*}