Weakly measurable function

Abstract: Assuming the continuum hypothesis, we construct a universally weakly measurable function from [0,1] into a dual of some weakly compactly generated Banach space, which is not Pettis integrable. This (partially) solves a problem posed by Riddle, Saab and Uhl [13]. We prove two results related to Pettis integration in dual Banach spaces. We also contribute to the problem whether it is consistent that every bounded function which is weakly measurable with respect to some Radon measure is Pettis integrable.

Abstract: This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

Abstract: Suppose $${\mathcal {X}}$$ is a real Banach space and $$(\varOmega , \varSigma , \mu )$$ is a probability space. We characterize the countable additivity of Henstock–Dunford integrable functions taking values in $${\mathcal {X}}$$ as those weakly measurable function $$ g: \varOmega \rightarrow {\mathcal {X}}, $$ for which $$\{y^*g: y^* \in B_{\mathcal {X}}^* \} $$ is relatively weakly compact in some separable Henstock–Orlicz space (in briefly H–Orlicz space) $$ H^{\theta }(\mu )$$ , where $$B_{\mathcal {X}}^*$$ is the closed unit ball in $${\mathcal {X}}^{*}.$$ We find relatively weakly compactness of some H–Orlicz space of Henstock–Gel’fand integrable functions.

Abstract: It is proved that a weakly compact generated Frechet space is Lindelof in the weak topology. As a corollary it is proved that for a finite measure space every weakly measurable function into a weakly compactly generated Frechet space is weakly equivalent to a strongly measurable function.