Polyhomomorphism of locally compact groups

Let $G, H$ be locally compact groups equipped with fixed two-side invariant Haar measures. A polyhomomorphism $G\to H$ is a pair
– subgroup $\Gamma$ in the product $G\times H$,
– a Haar measure on $\Gamma$ whose projections to $G$ and $H$ are dominated by the Haar measures on $G$ and $H$.
The set of all polyhomomorphisms $G\to H$ is a compact set with respect to the Shabauty topology. Polyhomomorphisms form a category, i.e., there is a well-defined product of polyhomomorphisms $G_1\to G_2$ and $G_2\to G_3$. We discuss some properties of such objects and an example (polyendomorphisms of locally compact linear spaces over finite fields).