Abstract of Colloquium talk at Math Department of WVU, September 4, 1997

Marek Balcerzak

Lodz Technical Univ, Poland

The behaviour of Borel and analytic sets under some operations

Let I be a sigma-ideal of subsets of a Polish space X. For a subset A of X^2 let Phi_I(A) denote the set of all x in X such the horizontal section A_x is not in I. We show that for certain sigma-ideals Phi_I sends Borel sets into Borel sets and analytic sets into analytic sets. Then let X=[0,1] and for a subset A of X^2 let D(A) (respectively, D_J(A)) denote the set of all pairs (x,y) in X^2 such that A_x is measurable (respectively, has the Baire property) and y is a density (respectively, category density) point of A_x. Mauldin observed that D sends Borel sets into Borel sets. We show that D sends analytic sets into analytic sets and that the analogous results hold for D_J.

The talk will be aimed at a general mathematical audience.