This question was asked in math stack exchange but received no replies:

https://math.stackexchange.com/questions/4000655/induced-representations-space-of-continuous-functions-on-g-to-a-hilbert-space

If this post violates community rules, please remark and I’ll remove it.

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:

- Why is $J$ (first yellow part) defined this way? why not just $(supp alpha)cap H$?
- 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):