**
Strong Fubini properties for measure and category
**

by

**
Krzysztof Ciesielski
and Miklos Laczkovich
**

*Fund. Math. 178(2)* (2003), 171-188.

Let (FP) abbreviate the statement that the integral equation

\int(\int f dy) dx = \int(\int f dx) dy
holds for every bounded function f:[0,1]^{2}-->**R**
whenever each of the integrals involved exists. We shall denote by (SFP)
the statement that the equality above holds for every
bounded function f:[0,1]^{2}-->**R** having measurable sections.
It follows from well-known results that both of (FP) and (SFP) are independent
from the axioms of ZFC. We investigate the logical connections of these statements
with several other strong Fubini type properties of the ideal of null sets. In particular,
we establish the equivalence of (SFP) to the
nonexistence of certain sets with paradoxical properties, a phenomenon that
was already known for (FP).
We also give the category analogues of these statements and, whenever
possible, we try to put the statements in a setting of general ideals as
initiated by Reclaw and Zakrzewski.

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**Last modified October 24, 2003.**