decomposition of Haar measure and Fubini theorem

decomposition of Haar measure and Fubini theorem

Let $G$ have a unique decomposition $G=AB$, where $G,A,B$ are linear Lie
groups with $G,A$ unimodular. Suppose we have Haar measure decomposition
$dg=dad_rb$ where $d_rb$ is the right Haar measure on $B$, that is, we
have
$$\int f(g)dg=\int f(ab) dad_rb$$
As far as I understand, $f$ has to be integrable over $G$, so that we
apply Fubini theorem and get the above formula, in which case we can
interchange the order of integration.
Now I'm curious how much the above formula depends on Fubini theorem?
Or, if we don't know $f$ is integrable, but we do have the right side
iterated integral convergent. Moreover, we may assume $f$ is smooth, the
function $\int f(ab)da$ is absolutely integrable as a function of $b$, do
we still have the above formula?
Thank you very much for any of your comment, and sorry for the question if
not appropriate here.