# Two notions of $G$-integrable topological vector spaces

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.