A function F:**R**^{2}-->**R** is *sup-measurable*
if F_{f}:**R**-->**R** given by F_{f}(x)=F(x,f(x)),
x in **R**, is measurable for each measurable function
f:**R**^{2}-->**R**. It is known that under different set theoretical
assumptions, including CH, there are sup-measurable non-measurable
functions, as well as their category analog.
In this paper we will show
that the existence of category analog of
sup-measurable non-measurable functions is independent of ZFC.
A similar result for the original measurable case
is the subject of a work in prepartion by
Roslanowski and Shelah.

**Last modified January 16, 2001.**