An analog of the Vitali-Hahn-Saks theorem

This paper considers $N$-triangular $s$-bounded set functions. We prove for these functions a fairly close analog both of the Vitali–Hahn–Saks theorem and of the corresponding results of Brooks and Darst for finitely additive vector measures. As simple corollaries, we obtain various modifications of the Vitali–Hahn–Saks theorem for certain classes of additive and nonadditive scalar and vector-valued set functions.