I wonder wether for a locally compact (for my purposes we'd also have unimodularity) group $G$ the L^p space $L^p(G)$ is a norm-closed subset of $C_b(G)$. The former space is of course meant to denote only those functions who are also bounded and continuous. In other words, the question is whether $L^p(G)\cap C_b(G)$ is norm closed in $C_b(G)$.
Thanks!
Best Answer
The answer depends on the measure you put on $G$. If the measure is finite, then the answer is affirmative, because if $f_n\in L^p(G)$ is a sequence converging in $C_b(G)$ then it converges uniformly and when the measure is finite this implies that the limit function is also in $L^p(G)$. However, if the measure is infinite (for example, the real line being your locally compact group with the Lebesgue measure) then there are examples of uniformly convergent sequences of integrable, continuous functions whose limit is not integrable.