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Adjusted wording in Haar.

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Bokuan Li
2026-06-20 10:13:19 -04:00
parent 094320ef29
commit ea091f006a
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src/measure/lcg/haar.tex
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@@ -119,7 +119,7 @@
Thus $\mathcal{I}(f) = \bracs{I_g(f)|g \in C_c^+(G) \setminus \bracs{0}}$ is precompact for each $f \in C_c^+(G)$. Thus $\mathcal{I}(f) = \bracs{I_g(f)|g \in C_c^+(G) \setminus \bracs{0}}$ is precompact for each $f \in C_c^+(G)$.
For each $V \in \cn_G(1)$, let $E_V = \bracs{I_g|g \in C_c^+(V) \setminus \bracs{0}}$, then $\fF = \bracs{E_V|V \in \cn_G(1)}$ is a filter on the product space $\prod_{f \in C_c^+(G)}\ol{\mathcal{I}(f)}$. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff}, $\bigcap_{V \in \cn_G(1)}\ol{E_V} \ne \emptyset$. For each $V \in \cn_G(1)$, let $E_V = \bracs{I_g|g \in C_c^+(V) \setminus \bracs{0}}$, then $\fF = \bracs{E_V|V \in \cn_G(1)}$ is a filter on the product space $\prod_{f \in C_c^+(G)}\ol{\mathcal{I}(f)}$. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff}, $\prod_{f \in C_c^+(G)}\ol{\mathcal{I}(f)}$ is compact, and $\bigcap_{V \in \cn_G(1)}\ol{E_V} \ne \emptyset$.
Let $I \in \bigcap_{V \in \cn_G(1)}\ol{E_V}$, then by continuity, Let $I \in \bigcap_{V \in \cn_G(1)}\ol{E_V}$, then by continuity,
\begin{enumerate}[label=(\roman*)] \begin{enumerate}[label=(\roman*)]