Proposition 27.2.8.label Let $G$ be a locally compact group, $\mu: \cb_{G} \to [0, \infty]$ be a left Haar measure, $p \in [1, \infty)$, and $E$ be a normed vector space over $K \in \RC$, then the mapping
\[G \times L^{p}(\mu; E) \quad (x, f) \mapsto L_{x}f\]
is jointly continuous. Similarly, if $\nu: \cb_{G} \to [0, \infty]$ is a right Haar measure, then
\[G \times L^{p}(\mu; E) \quad (x, f) \mapsto R_{x}f\]
is also jointly continuous.
Proof of the left case. Let $\eps > 0$, $x, y \in G$, and $f, g \in L^{p}(\mu; E)$, then
\begin{align*}\norm{L_xf - L_yg}_{L^p(\mu; E)}&\le \norm{L_xf - L_yf}_{L^p(\mu; E)}+ \norm{L_yf - L_y g}_{L^p(\mu; E)}\\&= \norm{L_xf - L_yf}_{L^p(\mu; E)}+ \norm{f - g}_{L^p(\mu; E)}\end{align*}
By Proposition 23.1.7, there exists $\phi \in C_{c}(G)$ such that $\norm{\phi - f}_{L^p(\mu; E)}< \eps$. In which case,
\begin{align*}\norm{L_xf - L_yf}_{L^p(\mu; E)}&\le \norm{L_xf - L_x \phi}_{L^p(\mu; E)}+ \norm{L_x\phi - L_y\phi}_{L^p(\mu; E)}\\&+ \norm{L_yf - L_y \phi}_{L^p(\mu; E)}\\&= 2\norm{f - \phi}_{L^p(\mu; E)}+ \norm{L_x\phi - L_y\phi}_{L^p(\mu; E)}\\&\le 2\eps + \normn{L_{x^{-1}y}\phi - \phi}_{u}\mu\bracs{\phi \ne 0}\end{align*}
By Proposition 27.1.2, there exists $V \in \cn_{G}(1)$ such that if $x^{-1}y \in V$, then $\norm{L_{x^{-1}y}\phi - \phi}_{u} < \eps/\mu\bracs{\phi \ne 0}$. Thus if $x^{-1}y \in V$, then
\[\norm{L_xf - L_yf}_{L^p(\mu; E)}\le 3\eps + \norm{f - g}_{L^p(\mu; E)}\]
$\square$
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