Title: Measurability of functions with approximately continuous
vertical sections and measurable horizontal sections
Authors: M. Laczkovich and Arnold W. Miller
Abstract: A function f:R -> R is approximately continuous iff
it is continuous in the density topology, i.e., for any ordinary
open set U the set E=f^{-1}(U) is measurable and has Lebesgue
density one at each of its points. Denjoy proved that
approximately continuous functions are Baire 1., i.e.,
pointwise limits of continuous functions.
For any f:R^2 -> R define f_x(y) = f^y(x) = f(x,y).
A function f:R^2 -> R is separately continuous if f_x and f^y
are continuous for every x,y in R. Lebesgue in his first paper
proved that any separately continuous function is Baire 1.
Sierpinski showed that there exists a nonmeasurable f:R^2 -> R
which is separately Baire 1.
In this paper we prove:
Thm 1. Let f:R^2 -> R be such that f_x is approximately continuous
and f^y is Baire 1 for every x,y in R. Then f is Baire 2.
Thm 2. Suppose there exists a real-valued measurable cardinal.
Then for any function f:R^2 -> R and countable ordinal i, if f_x
is approximately continuous and f^y is Baire i for every x,y in
R, then f is Baire i+1 as a function of two variables.
Thm 3. (i) Suppose that R can be covered by omega_1 closed null
sets. Then there exists a nonmeasurable function f:R^2 -> R such
that f_x is approximately continuous and f^y is Baire 2 for every
x,y in R. (ii) Suppose that R can be covered by omega_1 null sets.
Then there exists a nonmeasurable function f:R^2 -> R such that f_x
is approximately continuous and f^y is Baire 3 for every x,y in R.
Thm 4. In the random real model for any function f:R^2 -> R if
f_x is approximately continuous and f^y is measurable for every
x,y in R, then f is measurable as a function of two variables.