Let $G$ be a group, and a $H$ a subgroup, and for $a,b \in G$ let $a\sim b$ if $aH = bH$. Suppose I have a measure $\mu$ on $G/\sim$, the left cosets of $H$, and suppose that $G$ is equipped with a sigma algebra $\Sigma_G$ that is closed under right multiplication by $H$ and whose projection is contained in the sigma algebra of $G/\sim$. How can I define a measure $\nu$ on $G$ that satisfies $$\nu(AH) = \mu(\pi_\sim(A)),\qquad A \in \Sigma_G,$$ where $\pi_\sim : G \to G/\sim$ is the projection map and $AH = \{ ah : a\in A, h\in H \}$?
In my application, $G = SO(d)$ and $H = \{h\in G : hv=v\}$ for some fixed $v\in S^{d-1}$. The left cosets of $H$ are isomorphic to $S^{d-1}$, on which $\mu$ is defined.