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Microscopic and strongly microscopic sets on the plane. Fubini theorem and Fubini property

Karasińska A., Wagner-Bojakowska E.

Demonstratio Mathematica

2014

Vol. 47, nr 3

581--594

In this paper, we introduce the notions of microscopic and strongly microscopic sets on the plane and obtain a result analogous to Fubini Theorem.

Properness without elementaricity

Shelah S.

Journal of Applied Analysis

2004

Vol. 10, nr 2

169-289

We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary sub-models of some (H(x),∈ ). This leads to forcing notions which are "reasonably" definable. We present two specific properties materializing this intuition: nep (non-elernentary properness) and snep (Souslin non-elementary properness) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). A major theme here is "preservation by iteration", but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.

On the Fubini theorem for the Pettis integral for bounded functions

Michalak A.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 1

1-14

We show that if X is a WGG Banach space and it does not contain any isomorphic copy of l1, then for every bounded Pettis integrable function f : [0, 1]^2 --> X* there exists a scalarly equivalent function for which the Fubini theorem for the Pettis integral holds. On the other hand, we show that for every bounded Pettis integrable function f : [0, 1]^2 --> l^2 (R) there exists a scalarly equivalent bounded function for which the Fubini theorem for the Pettis integral does not hold. We also show (assuming the Martin axiom) that there exists a bounded Pettis integrable function f : [0, 1]^2 --> L^[infinity](lambda) such that for each function g scalarly equivalent to f the function s --> g(t, s) is not weakly measurable for almost every t [belongs to] [0, 1].