Lp spaces on groups

Remarks

Another Haar measure $c\mu$ gives the same space with the norm multiplied by $c^{1/p}$, so $L^p(G)$ does not depend on the choice of $\mu$ up to isometric scaling. Each $L_y$ is a linear isometry of $L^p(G)$, since $\mu$ is left-invariant. For $1 \le p < \infty$ and $f \in L^p(G)$, the map $y \mapsto L_y f$ from $G$ to $L^p(G)$ is continuous. For groups that are not $\sigma$-compact, $L^\infty(G)$ is usually defined modulo locally null sets, so that it is the dual of $L^1(G)$.

Example: $G = \Z$ with counting measure, a Haar measure. Then $L^p(\Z) = \ell^p(\Z)$, and $L_m$ shifts a sequence: $(L_m a)_n = a_{n - m}$, which keeps $\sum_n \abs{a_n}^p$.

Non-example of continuity for $p = \infty$: let $f = \mathbf{1}_{[0, \infty)}$ on $\R$. For $y > 0$, $L_y f - f = -\mathbf{1}_{[0, y)}$, so $\norm{L_y f - f}_\infty = 1$ for every $y > 0$, and $L_y f$ does not tend to $f$ as $y \to 0$.