Existence of disintegrations for improper priors on locally-compact groups

Question

In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically infinite Haar measures on locally-compact groups. In statistics, these are considered improper priors, which can frequently result in posteriors which are probability measures. However, this is not guaranteed, and is generally a thorny endeavor, so I am trying to tread carefully. See Chang & Pollard 1997 for examples.

Let $\eta$ be an infinite Haar measure on a locally-compact group $G$. Consider a measurable parameter set $\Theta$ which $G$ acts measurably on, and consider a random variable $\gamma : \Theta \to G$. For each $\theta$, define $k_\gamma^\theta(g) := \gamma(g^{-1} \theta)^{-1} g$, and the push-forward measures $\kappa_\gamma^\theta := \eta \circ (k_\gamma^\theta)^{-1}$ on $G$, which are infinite when $\eta$ is infinite.

If we consider a weighted version of $\eta$, i.e., $\alpha \eta$ for integrable $\alpha$, then a disintegration of $\alpha \eta$ through $k_\gamma^\theta$ exists by the disintegration theorem, since $\alpha \eta$ is Radon. However, I don't believe this holds for the infinite measure $\eta$.

Does a disintegration exist for the infinite measure $\eta$? If not, then what additional assumptions must we make for there to exist a disintegration of $\eta$ through $k_\gamma^\theta$?

Answer 1

Ok I think I've got it, following the partition argument indicated by Michael Greinecker in the comments above. Thank you Michael! Please let me know if you spot any errors.

Lemma. Let $G$ be a locally compact group with Borel $\sigma$-algebra $\mathcal{B}(G)$. For each measurable function $k : G \to G$ and the push-forward measure $\kappa := k_* \eta = \eta \circ k^{-1}$ on $G$, there exists a disintegration $\eta_k(\mathrm{d} g|g')$ such that for any $B \in \mathcal{B}(G)$, \begin{equation} \eta(B) = \int_G \eta_k(B|g') \kappa(\mathrm{d} g') = \int_G \eta_k(B|k(g)) \eta(\mathrm{d} g). \end{equation}

Proof. We construct the global disintegration by a local partition argument. By local compactness of $G$, there exists a countable partition $\mathcal{C}$ of disjoint pre-compact sets of finite Haar measure, whose union equals $G$.

For each $C \in \mathcal{C}$, define the probability measure $\eta^C := \frac{1}{\eta(C)} 1_C \eta$. Each $\eta^C$ is Radon, so by the disintegration theorem (cf. Theorem 3.1 of Leao et al. 2004), there exists a regular conditional probability through $k$, i.e., a measurable measure-valued function $g' \mapsto \eta^C_k(\mathrm{d} g|g')$ such that the disintegration equation holds for each $B \in \mathcal{B}(G)$: \begin{equation} \frac{\eta(C \cap B)}{\eta(C)} = \eta^C(B) = \int_G \eta^C_k(B|g') \kappa(\mathrm{d} g') = \int_G \eta_k^C(B|k(g)) \eta(\mathrm{d} g). \end{equation}

We now define $\eta_k$ by combining across partition sets. Suppose $B \in \mathcal{B}(G)$ has finite Haar measure $\eta(B) < \infty$. Then: \begin{equation} \eta(B) = \sum_C \eta(C \cap B) = \int_G \sum_{C \in \mathcal{C}} \eta(C) \eta^C_k(B|g') \kappa(\mathrm{d} g') =: \int_G \eta_k(B|g') \kappa(\mathrm{d} g'), \end{equation} where we define $\eta_k(B|g') := \sum_{C \in \mathcal{C}} \eta(C) \eta^C_k(B|g')$.