On the theory of vector measures

Topological Properties of the Complex Vector Lattice of Bounded Measurable Functions

“…In this section we study the Mackey topology ( (Σ), (Σ)) on (Σ). It is known that ( (Σ), ( (Σ), (Σ))) is a generalized DF-space, that is, ( (Σ), (Σ)) is the finest locally convex topology agreeing with itself on norm-bounded sets in (Σ) (see [8,Corollary 11.8] (iv)⇒(i) Assume that ∈ (Σ). Let ( ) be a sequence in…”

Classical extensions of the ring of continuous functions and the corresponding preimages of a completely regular space

“…The Borel preimage was studied in [46,48,49]. The concrete character of the realizing morphism is established by the following Proposition.…”

“…According to the criterion from 1.3.2, this a-preimage exists and is a realizing asL-preimage of the crLextension u: C ~-+ C ~'. This preimage was examined in [32,34,35,39,46,48,49,76]. Since the hyperstonean preimage is defined via C t', Fremlin, in his review of the monograph [59], raised the problem of the ez.istence of a direct topological construction of the hyperstonean preimage T ~ hT out closed sets and, correspondingly, zero-sets of a space T. A function f: T ~ R will be called an A-function if for any n there exists a finite covering {A,,i I i} C A of the space T such that w(f, …”

[Weakly] compact operators and DF spaces

“…A further concrete situation for which Theorems 2.2 and 2.3 provide new tools, is Graves' [13] "linearization of vector measures": For a σ-algebra Σ of subsets of a set S and a Banach space X, the space of bounded vector measures from Σ into X is in one-to-one correspondence with the continuous linear operators from the space S^(Σ) of J-simple functions, endowed with the sup-norm topology, into X: μ H* integration with respect to μ. W. H. Graves in [13] specified an lc topology τ on £^(Σ), coarser than the sup-norm topology, which singles out the strongly countably additive vector measures as exactly those whose associated operators are τ-continuous. (£^(Σ), τ) (1) There exists a zero neighbourhood U in X and a sequence (C n ) neN of ^-precompact (rest, p-weakly relatively compact) disks in Y such that H(U) c Π {C n + V n \ n e N}, where (F n ) ne jv is a (decreasing) neighbourhood base in Y.…”

“…(Consequences of this observation for such function spaces have been the point of discussion of the paper [33]; see also the survey [34]. ) Further examples in this context are F. D. Sentilles' [41] strict topology β on L°° in his L°°-ZΛ-duality, and the "universal strongly countably additive" topology τ on the space £^(&) (of simple functions on a ring <3% of subsets of a set S) of W. H. Graves [13] …”