Let \(G\) be a locally compact group. A crucial lemma in this article, of Hahn-Banach type, shows that if \(G\) is amenable and equipped with a weight, i.e., a measurable submultiplicative positive function \(\omega\) on \(G\), existence of an invariant mean on \(L^ \infty(G)\) implies existence of a continuous positive character \(\Phi\) on \(G\) such that \(\Phi\leq\omega\). Making use of a result due to \textit{N. Grønbaek} [Trans. Am. Math. Soc. 319, No.\ 2, 765-775 (1990; Zbl 0701.46035)], the author then can prove that if \(\omega\) is a continuous normalized weight on \(G\) and the Banach algebra \(L^ 1(G,\omega)\) of functions on \(G\) that are integrable with respect to \(\omega\) is an amenable algebra, there exists a positive character \(\Phi\) on \(G\) inducing an isomorphism of \(L^ 1(G,\omega)\) onto \(L^ 1(G)\) via the mapping \(\varphi\mapsto\Phi(g)\varphi\). Hahn-Banach type extensions of characters from a closed invariant normal subgroup to the entire group are established. Counterexamples elucidate the necessity of amenability assumptions.