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Strict topologies as topological algebras

Surjit Singh Khurana — 2001

Czechoslovak Mathematical Journal

Let X be a completely regular Hausdorff space, C b ( X ) the space of all scalar-valued bounded continuous functions on X with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally m -convex.

Positive vector measures with given marginals

Surjit Singh Khurana — 2006

Czechoslovak Mathematical Journal

Suppose E is an ordered locally convex space, X 1 and X 2 Hausdorff completely regular spaces and Q a uniformly bounded, convex and closed subset of M t + ( X 1 × X 2 , E ) . For i = 1 , 2 , let μ i M t + ( X i , E ) . Then, under some topological and order conditions on E , necessary and sufficient conditions are established for the existence of an element in Q , having marginals μ 1 and μ 2 .

Product of vector measures on topological spaces

Surjit Singh Khurana — 2008

Commentationes Mathematicae Universitatis Carolinae

For i = ( 1 , 2 ) , let X i be completely regular Hausdorff spaces, E i quasi-complete locally convex spaces, E = E 1 ˘ E 2 , the completion of the their injective tensor product, C b ( X i ) the spaces of all bounded, scalar-valued continuous functions on X i , and μ i E i -valued Baire measures on X i . Under certain conditions we determine the existence of the E -valued product measure μ 1 μ 2 and prove some properties of these measures.

Radon-Nikodym property

Surjit Singh Khurana — 2017

Commentationes Mathematicae Universitatis Carolinae

For a Banach space E and a probability space ( X , 𝒜 , λ ) , a new proof is given that a measure μ : 𝒜 E , with μ λ , has RN derivative with respect to λ iff there is a compact or a weakly compact C E such that | μ | C : 𝒜 [ 0 , ] is a finite valued countably additive measure. Here we define | μ | C ( A ) = sup { k | μ ( A k ) , f k | } where { A k } is a finite disjoint collection of elements from 𝒜 , each contained in A , and { f k } E ' satisfies sup k | f k ( C ) | 1 . Then the result is extended to the case when E is a Frechet space.

Lattice-valued Borel measures. III.

Surjit Singh Khurana — 2008

Archivum Mathematicum

Let X be a completely regular T 1 space, E a boundedly complete vector lattice, C ( X ) ( C b ( X ) ) the space of all (all, bounded), real-valued continuous functions on X . In order convergence, we consider E -valued, order-bounded, σ -additive, τ -additive, and tight measures on X and prove some order-theoretic and topological properties of these measures. Also for an order-bounded, E -valued (for some special E ) linear map on C ( X ) , a measure representation result is proved. In case E n * separates the points...

Order convergence of vector measures on topological spaces

Surjit Singh Khurana — 2008

Mathematica Bohemica

Let X be a completely regular Hausdorff space, E a boundedly complete vector lattice, C b ( X ) the space of all, bounded, real-valued continuous functions on X , the algebra generated by the zero-sets of X , and μ C b ( X ) E a positive linear map. First we give a new proof that μ extends to a unique, finitely additive measure μ E + such that ν is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of E + -valued finitely additive measures on are proved, which extend...

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