Finitely additive representation of \(L^{p}\) spaces (Q879033)

Let \(B\) denote the \(\sigma\)-algebra of Borel subsets of \(\{0,1\}^N\), let \(\mu\) be an atomless countably additive probability on \(B\), let \(T\) be a countable dense subset of \(\{0,1\}^N\), and let \(\Sigma\) denote its power set. The author shows that there is a purely finite additive probability \(\lambda\) on \(\Sigma\) such that, for \(1\leq p<\infty\), there exists a unique isometric isomorphism between \(L^p(\mu)\) and a closed linear subspace of \(L^p(\lambda)\) mapping simple and continuous functions on \(\{0,1\}^N\) to their corresponding restrictions to \(T\).