# 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):