A common generalization of Kelley's theorem (concerning measures on Boolean algebra) and on von Neumann's minimax theorem
A new proof of Kelley's Theorem
Kelley's Theorem is a purely combinatorial characterization of measure algebras. We first apply linear programming to exhibit the duality between measures and this characterization for finite algebras. Then we give a new proof of the Theorem using methods from nonstandard analysis.
A noncommutative version of a Theorem of Marczewski for submeasures
It is shown that every monocompact submeasure on an orthomodular poset is order continuous. From this generalization of the classical Marczewski Theorem, several results of commutative Measure Theory are derived and unified.
A Note on the Extension of Measures on Lattices
A note on the product of measure algebras
A problem of von Neumann and Maharam about algebras supporting continuous submeasures
We show that a σ-algebra 𝔹 carries a strictly positive continuous submeasure if and only if 𝔹 is weakly distributive and it satisfies the σ-finite chain condition of Horn and Tarski.
A Simple Example of Pathological Submeasure.
Almost every tree function is independent
An algebraic and logical approach to continuous images
Bands of invariantly extensible measures.
Boolean operations over measure algebras
Borel extensions of Baire measures in ZFC
We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.
Borel semigroups and probabilities.
Continuity and Extension of Valuations
Convergence almost everywhere of sequences of measurable functions
Convergence and submeasures in Boolean algebras
A Boolean algebra carries a strictly positive exhaustive submeasure if and only if it has a sequential topology that is uniformly Fréchet.
Dedekind extension of measures with values in vector lattices.
Dividing measures and narrow operators
We use a new technique of measures on Boolean algebras to investigate narrow operators on vector lattices. First we prove that, under mild assumptions, every finite rank operator is strictly narrow (before it was known that such operators are narrow). Then we show that every order continuous operator from an atomless vector lattice to a purely atomic one is order narrow. This explains in what sense the vector lattice structure of an atomless vector lattice given by an unconditional basis is far...
Duality for Cardinal Algebras.
Examples of ε-exhaustive pathological submeasures
For any given ε > 0 we construct an ε-exhaustive normalized pathological submeasure. To this end we use potentially exhaustive submeasures and barriers of finite subsets of ℕ.