Measures on topological spaces

Locally Compact Spaces of Measures

“…terminology follow quite closely those in the paper of Varadarajan [5]. In particular, our topological definitions and terminology coincide with those of Kelley [3].…”

“…Varadarajan [5,Remark,p. 205] gives conditions under which the hypotheses of Theorem 5 are fulfilled.…”

“…Using the methods and results of Varadarajan [5], we give necessary and sufficient conditions on the topological space X for the local compactness of the spaces of ff-smooth, rsmooth, and tight measures [resp., signed measures] on X, respectively, when endowed with the weak topology. For such spaces of signed measures, the finiteness of X is easily seen to be necessary and sufficient (Remark 4).…”

A Nonstandard Characterization of Weak Convergence

“…It is well known [11,Theorem 6,p. 165] that, if <p E CiX)d, there is a finitely additive measure v¿.…”

Compactness and sequential completeness in some spaces of operators

“…Then M(X ) with the norm μ = |μ|(X ) (= the total variation of μ) and the natural order (μ 1 ≤ μ 2 if μ 1 (A) ≤ μ 2 (A) for all A ∈ B) is a Dedekind complete Banach lattice (see [20, p. 114, p. 122]). Due to the Alexandrov representation theorem (see [19], [20,Theorem 5.1]) C b (X ) can be identified with M(X ) through the lattice isomorphism M(X ) μ → μ ∈ C b (X ) , where μ (u) = X udμ for all u ∈ C b (X ), and μ = μ . The strict topologies β σ , β τ and β t on C b (X ) are of importance in the topological measure theory (see [18], [20] for more details).…”