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https://math.stackexchange.com/questions/4000655/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, $q:G\rightarrow G/H$ 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 $\left\Vert u\right\Vert _{\sigma}$ and $\left\langle u,v\right\rangle _{\sigma}$, and we denote by $C(G,\mathcal{H}_{\sigma})$ the space of continuous functions from $G$ to $\mathcal{H}_{\sigma}$.  

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

> **Proposition.** If $\alpha:G\rightarrow\mathcal{H}_{\sigma}$ is continuous with compact support, then the function
> $f_{\alpha}(x)=\int_{H}\sigma(\eta)\alpha(x\eta)d\eta$ 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 $\alpha\in C_{c}(G,\mathcal{H}_{\sigma})$.

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][1]][1]  

  [1]: https://i.sstatic.net/g9sSE.png