Haar Measure on Semidirect Product of Unimodular Groups Let $M, N$ be closed subgroups of a given locally compact topological group such that the product $P = MN$ is semidirect, with $M$ normalizing $N$.  Assume second countable, sigma compact, whatever so that Radon products coincide with product measures.
To make life easy, assume $M, N$ are unimodular.  How do we construct a Haar measure on $P$?
 A: Let $\mu$ be a Haar measure on $N$.  For $m \in M$, the map $\textrm{Int } m$ given by $g \mapsto mgm^{-1}$ normalizes $N$, and $\mu \circ \textrm{Int } m$ defines a Haar measure on $N$, where we identify $\textrm{Int } m$ with a bijection on the Borel sets of $N$.  Hence there exists a constant $\delta(m) > 0$ such that $\delta(m) \mu = \mu \circ \textrm{Int } m$.  Therefore, for $f \in C_c(N)$,
$$\int\limits_N f(mnm^{-1})dn = \int\limits_N f \circ \textrm{Int } m \, d\mu  = \int\limits_N f \, d(\mu \circ \textrm{Int } m^{-1})$$ $$ = \delta(m^{-1}) \int\limits_N f \, d\mu = \delta(m^{-1})\int\limits_N f(n)dn$$
where the second equality is a change of variables formula (e.g. see Haar measure scalars, what am I doing wrong here?).  Also, it's easy to check that $\delta: M \rightarrow (0,\infty)$ is a homomorphism.  It's also continuous.
As topological spaces, we can identify $P = M \times N$, and by the Riesz representation theorem, we get a Radon measure on $P$ from the positive linear functional on $C_c(P)$ by sending $f \in C_c(P)$ to $\int\limits_N \int\limits_M f(mn)\delta(m)dmdn$.  To say that this Radon measure is a right Haar measure is to say that this linear functional is invariant on the right.
It is clearly invariant on the right by $N$.  To checking finish invariance, let $m_0 \in M$.  We have 
$$\int\limits_N \int\limits_M f(mnm_0)\delta(m)dmdn = \int\limits_N \int\limits_M f(mm_0m_0^{-1}nm_0)\delta(m)dmdn$$
$$ = \delta(m_0) \int\limits_N \int\limits_M f(mm_0n)\delta(m)dmdn  = \int\limits_N \int\limits_M f(mn)\delta(m)dmdn$$
where in the last equality we use the fact that $\delta$ is a homomorphism and make the change of variables $m \mapsto mm_0^{-1}$.
To get a left Haar measure on $P$, you use the linear functional $f \mapsto \int\limits_M \int\limits_N f(nm) \delta(m^{-1}) \, dn dm$
A: For future reference, Prop.s 28 & 29 on pp. 99-100 of Nachbin's The Haar Integral address exactly this problem (without the unimodularity assumption).
