An infinite dimensional purification principle without saturation

A bang-bang principle for the conditional expectation vector measure

“…Proof. In this proof, some arguments dealing mainly with measurability considerations are taken from [2].…”

“…The previous theorem works by assuming instead of ( 5), that G < < µ, see for instance [2], Remark 1.…”

“…Lemma 1 (Lemma 2 in [2]). Let X a Banach space and G : A → X be a Lyapunov vector measure and α i : Ω → [0, 1], i = 1, ..., n, be measurable functions, such that…”

“…Generally, when the range space of the vector measure is infinite dimensional, the classical convexity theorem of Lyapunov fails. However, an infinite dimensional version of such result established by Knowles [29] can be used to establish a purification principle as in Askoura [2]. Furthermore, many works succeeded to recover this property and to establish adequate bang-bang and purification results by using Maharam types and saturated measure spaces; see Greinecker and Podczeck [10], Khan and Sagara [22,23,24,25], Sagara [32] for infinite dimensional Lyapunov convexity theorems, purification processes and applications to the integration of set-valued mappings, equilibrium theory and control systems.…”