New characterizations of some \(L^p\)-spaces (Q1578334)

Let \((X,\Sigma ,\mu)\) be a complete measure space. Some new characterizations of such positive measure \(\mu \) for which \(L^{p}(X,\mu) \subset L^{q}(X,\mu)\), \(0\prec p\prec q,\) are stated. Furthermore, six equivalent conditions are given which force \(L^{p}(X,\mu)\), \(1\leq p\prec \infty ,\) to be isometrically isomorphic to \(l^{p}(\Gamma)\) for some index set \(\Gamma \) which depends only on \((X,\Sigma ,\mu)\).