diff --git a/src/measure/lcg/haar.tex b/src/measure/lcg/haar.tex index e68d3dc..2a1c23d 100644 --- a/src/measure/lcg/haar.tex +++ b/src/measure/lcg/haar.tex @@ -45,7 +45,7 @@ Let $G$ be a locally compact group and $f, h, g \in C_c^+(G)$, then: \begin{enumerate} \item If $g \ne 0$, then $(f: g) < \infty$. - \item $(f, h: g) \le (h: g) + (h: g)$. + \item $(f + h: g) \le (f: g) + (h: g)$. \item For each $\lambda \ge 0$, $(\lambda f: g) = \lambda(f: g)$. \item If $f \le h$, then $(f: g) \le (h: g)$. \item $(f: g) \le (f: h)(h: g)$. @@ -126,7 +126,7 @@ \item For each $f \in C_c^+(G) \setminus \bracs{0}$, $I(f) \in [(h: f)^{-1}, (f: h)]$. \item For every $\lambda \ge 0$ and $f \in C_c^+(G)$, $I(\lambda f) = \lambda I(f)$. \item For any $x \in G$, $I(L_xf) = I(f)$. - \item For each $f, f' \in C_c^+(G)$, $I(f + g) \le I(f) + I(g)$. + \item For each $f, f' \in C_c^+(G)$, $I(f + f') \le I(f) + I(f')$. \end{enumerate} Let $f, f' \in C_c^+(G)$ and $\eps > 0$. By \autoref{lemma:haar-approx}, there exists $V \in \cn_G(1)$ such that for each $g \in E_V$, @@ -137,7 +137,7 @@ In which case, $I(f) + I(f') \le I(f + f') + 3\eps$. Since this holds for all $\eps > 0$, \begin{enumerate}[start=4, label=(\roman*)] - \item For each $f, f' \in C_c^+(G)$, $I(f + g) \ge I(f) + I(g)$. + \item For each $f, f' \in C_c^+(G)$, $I(f + f') \ge I(f) + I(f')$. \end{enumerate} Using \autoref{lemma:positive-functional-extension}, $I$ extends to a positive linear functional on $C_c(G; \real)$, with $I(f) > 0$ for all $f \in C_c^+(G) \setminus \bracs{0}$, and