In this paper we present various weak star Kuratowski convergence results for multivalued martingales, supermartingales and multivalued mils in the dual of a separable Banach space. We establish several integral representation formulas for convex weak star compact valued multifunctions defined on a Köthe space and derive several existence results of conditional expectation for multivalued Gelfand-integrable multifunctions. Similar convergence results for Gelfand-integrable martingales in the dual...

Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $\lambda \mathbb{P}\left(S\right(f\left)\ge \lambda \right)\le \pi /2\parallel f\parallel ₁$, and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.