Universal summands for families of measurable functions

by

Tomasz Natkaniec and Ireneusz Reclaw

10 pages

(1) For any countable ordinal number \alpha there exists a Borel measurable function g:R-->R such that g+f is a Darboux function (is almost continuous in the sense of Stallings) for every f in $B$\alpha. This solves a problem of J. Ceder.

(2) There is a function g that is universally measurable and has the Baire property in restricted sense such that g+f is Darboux for every Borel measurable function f.

(3) There is g:R-->R such that f+g is extendable for each f:R-->R that is Lebesgue measurable (has the Baire property).

(4) For every countable ordinal number \alpha, each f from $B$\alpha is the sum of two extendable functions g and h from $B$\alpha. This answers a question of A. Maliszewski.

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