A functional characterization of measures and the Banach-Ulam problem (Q607317)

The first main result in this paper is that a real-valued finitely additive measure \(\mu\) on a measurable space \((\Omega,\mathcal A)\) is countably additive if and only if the corresponding functional \(\phi_\mu(x):=\int_\Omega x \, d\mu\) is sequentially continuous for the weak\({}^*\) topology on \(\ell_\infty (\mathcal A)\). Then, motivated by the Yosida-Hewitt desomposition theorem of finitely additive measures into a countably additive part plus a purely finitely additive part, the authors introduce the notions of purely weak\({}^*\)-sequential continuity and purely strong continuity functionals (in the duals of subspaces of dual Banach spaces) to prove that every continuous functional on \(\ell_\infty(\mathcal A)\) can be uniquely decomposed into the \(\ell_1\)-sum of a weak\({}^*\)-continuous part, a purely weak\({}^*\)-sequentially continuous part and purely strongly continuous part. This second main result is then used to give several applications to measure extensions.