The composition of operator-valued measurable functions is measurable

Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology

Let X and Y be Banach spaces and (Ω,Σ, μ) a finite measure space. In this note we introduce the space L[μ;L (X, Y )] consisting of all (equivalence classes of) functions Φ : Ω 7→ L (X, Y ) such that ω 7→ Φ(ω)x is strongly μ-measurable for all x ∈ X and ω 7→ Φ(ω)f(ω) belongs to L(μ; Y ) for all f ∈ L ′ (μ;X), 1/p + 1/p = 1. We show that functions in L[μ;L (X, Y )] define operator-valued measures...

Generalized Rings of Measurable and Continuous Functions

This paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

When is the ring of real measurable functions a hereditary ring?

‎Let $M(X‎, ‎mathcal{A}‎, ‎mu)$ be the ring of real-valued measurable functions‎ ‎on a measure space $(X‎, ‎mathcal{A}‎, ‎mu)$‎. ‎In this paper‎, ‎we characterize the maximal ideals in the rings of real measurable functions‎ ‎and as a consequence‎, ‎we determine when $M(X‎, ‎mathcal{A}‎, ‎mu)$ is a hereditary ring.

On Spaces of Measurable Vector - Valued Functions 399 Proof

Towers of Measurable Functions

We formulate variants of the cardinals f, p and t in terms of families of measurable functions, in order to examine the effect upon these cardinals of adding one random real.