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In 1940, Lyapunov proved that th€ range ofn finite measur€s on the same measurable spa.e (O,"4) is compa.t and if the measures a.re atoDless theD the range i6 also conv€x. There har€ b€€n many ext€trsioEs of Lyapunov's convexiry theorem; for example se€ Dubins and SpaDier [3] ad Droretzky, Wsld and Wolfos'itz [4] . This paper presents s new geDeralization of this theorem which i6 of speciÄl interest in th€ theory of optimal stopping. The sa.me conctuions hold uder the wea,kened assumptioDs of havina adapted a-algebras ft e fz q . q .F", wherc { (O, fr, p; )}i= r is a fniie collection of 6nitc meäsure spac€6 on the sa.me uaderlying set 0. Convexity of the range for geDeräI (not necessärily nested) r-algebrss Fr,. . . ,f. may fait (Example 2.3 in 16l).

A generalization of Lyapunov’s convexity theorem with applications in optimal stopping