Can somebody put me in the right direction to prove that:
$\lim_{p \to 1} \lVert f \rVert_{p}^p=\lVert f \rVert_{1}$ ?
Maybe this will be a beginning:
If $f \in$ $\mathcal{L}^1(\mu)\cap \mathcal{L}^2(\mu)$, it follows that $f \in \mathcal{L}^p(\mu)$ for all $1\leq p\leq 2$.
Thanx in advance!
Prove that $p \mapsto \|f\|_p$ is continuous on $[1,2]$ and then by continuity of $t \mapsto t^p$ we know that $p \mapsto \|f\|_p^p$ is continuous on $[1,2]$.
It follows that $$\lim_{p \to 1} \lVert f \rVert_{p}^p=\lVert f \rVert_{1}$$