In Part I of this report the pointwise derivation of scalar set functions is investigated, first along the lines of R. DE POSSEL (abstract derivation basis) and A. P. MORSE (blankets); later certain concrete situations (e. g. , the interval basis) are studied. The principal tool is a Vitali property, whose precise form depends on the derivation property studied. The "halo" (defined at the beginning of Part I, Ch. IV) properties can serve to establish a Vitali property, or sometimes produce directly a derivation property. The main results established are the theorem of JESSEN-MARCINKIEWICZ-ZYGMUND (Part I, Ch. V) and the theorem of A. P. MORSE on the universal derivability of star blankets (Ch. VI) . . In Part II, points are at first discarded; the setting is somatic. It opens by treating an increasing stochastic basis with directed index sets (Th. I. 3) on which premartingales, semimartingales and martingales are defined. Convergence theorems, due largely to K. KRICKEBERG, are obtained using various types of convergence: stochastic, in the mean, in Lp-spaces, in ORLICZ spaces, and according to the order relation. We may mention in particular Th. II. 4. 7 on the stochastic convergence of a submartingale of bounded variation. To each theorem for martingales and semi-martingales there corresponds a theorem in the atomic case in the theory of cell (abstract interval) functions. The derivates concerned are global. Finally, in Ch
CONTENT
I Pointwise Derivation -- I: Derivation Bases -- II: Derivation Theorems for ?-additive Set Functions under Assumptions of the Vitali Type -- III: The Converse Problem I: Covering Properties Deduced from Derivation Properties of ?-additive Set Functions -- IV: Halo Assumptions in Derivation Theory. Converse Problem II -- V: The Interval Basis. The Theorem of Jessen-Marcin-Kiewicz-Zygmund -- VI: A. P. Morseโ{128}{153}s Blankets -- II Martingales and Cell Functions -- I: Theory without an Intervening Measure -- II: Theory in a Measure Space without Vitali Conditions -- III: Theory in a Measure Space with Vitali Conditions -- IV: Applications -- Complements -- 1ยฐ. Derivation of vector-valued integrals -- 2ยฐ. Functional derivatives -- 3ยฐ. Topologies generated by measures -- 4ยฐ. Vitaliโ{128}{153}s theorem for invariant measures -- 5ยฐ. Global derivatives in locally compact topological groups. -- 6ยฐ. Submartingales with decreasing stochastic bases -- 7ยฐ. Vector-valued martingales and derivation -- 9ยฐ. Derivation of measures
