I've been working through Hochschild and Mostow's paper and they have the following definition of a $G$-integrable topological vector space $A$ where $G$ is a locally compact topological group. Below, Let $F^*(G, A)$ denotes the space of continuous functions $G\to A$, topologized with the compact open topology.
$A$ is $G$-integrable if there is a continuous map $J:F^*(G, A)\to A$ and a separating family $A'$ of continuous linear functionals on $A$ such that for every $\gamma\in A'$ and $f\in F^*(G, A)$ we have $$\gamma(J(f))=\int_G \gamma(f(g))d\mu(g)$$ where $\mu$ is the Haar measure of $G$.$A$ is a $G$-integrable continuous $G$-module if it is $G$-integrable with respect to a separating family $A'$ having the property that if $\gamma\in A'$ then for every $g\in G$ the function $a\mapsto \gamma(ga)$ is in $A'$.
In an attempt to get some sort of intuition about this (to me at least) fairly obtuse definition, I came across a similar definition of $G$-integrability for $A$ a Hilbert space and $G$ a locally compact unimodular group acting on $A$ by unitary transformations:
A unitary representation $A$ is $G$-integrable if there is a non-zero $a\in A$ such that $\int_G(ga, a) d\mu(g)$ is finite.
This brings me to my question:
Question. Is there a relationship between these two notions, when $A$ is a unitary representation of $G$ and $G$ is a locally compact unimodular group?I would guess the separating family required is $A'=\{a\mapsto (b, a): b\in A\}$, but I can't see how to proceed.