Co-countable sets of uniqueness for series of independent random variables

Given a sequence of independent random variables (fk) on a standard Borel space Ω with probability measure μ and a measurable set F, the existence of a countable set S⊂F is shown, with the property that series k_∑ckfk which are constant on S are constant almost everywhere on F. As a consequence, if the functions fk are not constant almost everywhere, then there is a countable set S⊂Ω such that the only series k_∑ckfk which is null on S is the null series; moreover, if there exists b<1 such that for every k and every α, then the set S can be taken inside any measurable set F with μ(F)>b.     

The Bartle Bilinear Integration and Carleman Operators

Some characterizations of integrable functions in the bilinear sense of Bartle with respect to the injective tensor product are obtained. As a consequence it is shown that the kernels of Carleman compact operators coincide with these Bartle integrable functions. This result is applied to prove that every Carleman L-weak-compact operator is compact. An example showing the different behavior of the integrability with respect to the projective tensor product is given. A general Fubini theorem in this setting is shown.     

Tensor Products and Operators in Spaces of Analytic Functions

Let X be an infinite dimensional Banach space. The paper provesthe non-coincidence of the vector-valued Hardy space H^p(T, X)with neither the projective nor the injective tensor productof H^p(T) and X, for 1 < p < . The same result is provedfor some other subspaces of L^p. A characterization is givenof when every approximable operator from X into a Banach spaceof measurable functions F(S) is representable by a functionF:S X* as x F(·), x. As a consequence the existenceis proved of compact operators from X into H^p(T) (1 p <) which are not representable. An analytic Pettis integrablefunction F:T X is constructed whose Poisson integral does notconverge pointwise.