A Note on the Strong-Operator Topology or Countably Generated $O$-Vector Spaces

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...

A Strong Operator Topology Adiabatic Theorem

Finitely Generated Vector Spaces

But before starting on this, let me try to explain again, in a different way, our approach. The one habit I’ve been trying to wean you of is the an over-reliance upon concrete examples to develop your understanding. The vector space R is a very concrete and familar example of a vector space over a field. To do calculations in this setting all you need to do is apply arithmetic (over and over an...

A Note on the Strong Law of Large Numbers

Petrov (1996) proved the connection between general moment conditions and the applicability of the strong law of large numbers to a sequence of pairwise independent and identically distributed random variables. This note examines this connection to a sequence of pairwise negative quadrant dependent (NQD) and identically distributed random variables. As a consequence of the main theorem ...

On Σ-countably Tight Spaces

Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality c if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and σ-countably tight compactum has cardinality c remains open. We also show that if an arbitrary product is σ-countably tight then all but finitel...