The Gross–Sazonov theorem for sign-variable cylindrical measures

A topological condition is found whose fulfilment is sufficient for $\sigma$-additivity of bounded signed cylindrical measures. The topology $\tau_{\mathrm{C}\Gamma}$ used in this condition is in general strictly stronger than Sazonov's topology; however, $\tau_{\mathrm{C}\Gamma}$-continuity of the Fourier transformation of a cylindrical measure entails its $\tau_{\mathrm{C}}$-continujty.