rt.representation theory – Induced representations: space of continuous functions on $G$ to a Hilbert space

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Let $G$ be a locally compact group, $H$ a closed subgroup, $q:Grightarrow G/H$ the canonical quotient map, and $sigma$ a unitary representation of H on $mathcal{H}_{sigma}$. We denote the norm and inner product on $mathcal{H}_{sigma}$ by $leftVert urightVert _{sigma}$ and $leftlangle u,vrightrangle _{sigma}$, and we denote by $C(G,mathcal{H}_{sigma})$ the space of continuous functions from $G$ to $mathcal{H}_{sigma}$.

Let $mathcal{F}_{0}={ fin C(G,mathcal{H}_{sigma}):q(supp f)$ compact, $f(xxi)=sigma(xi^{-1})f(x)$ for $xin G,:xiin H}$.

Proposition. If $alpha:Grightarrowmathcal{H}_{sigma}$ is continuous with compact support, then the function
$f_{alpha}(x)=int_{H}sigma(eta)alpha(xeta)deta$ belongs to
$mathcal{F}_{0}$ and is left uniformly continuous on $G$. Moreover,
every element of $mathcal{F}_{0}$ is of the form $f_{alpha}$ for
some $alphain C_{c}(G,mathcal{H}_{sigma})$.

The following questions refer to the attached section below:

  1. Why is $J$ (first yellow part) defined this way? why not just $(supp alpha)cap H$?
  2. Regarding the second yellow part, it might be a silly question, but what if $|J|$ is infinity?

Attached is the relevant section of the proof from A Course in Abstract Harmonic Analysis by Gerald B. Folland, 2nd edition, prop 6.1 (Note that proposition 2.6, mentioned in the text, refers to the fact that $alpha$ is uniformly continuous):
enter image description here