Wikipedia says:
One property of a left Haar measure $\mu$ is that, letting $s$ be an element of $G$, the following is valid:
$$\int_G f(sx) d\mu(x) = \int_G f(x) d\mu(x)$$
for any Haar integrable function $f$ on $G$.
How does one generalize this property to $G$ being a locally compact Hausdorff matrix group, i. e. for the Bochner integral?
I have no idea how to show it, since the monotone convergence theorem does not apply for Bochner integrals. And yes, you can project: $$proj_{11}\left(\int_G f(sx) d\mu(x) \right) = \int_G proj_{11}(f(sx)) d\mu(x)$$ but this gives us not a Lebesgue integral nor a real-valued argument $x$. In addition, I have no idea where I could find such a proof.