There is no continuous projection from the measurable functions on the square onto the measurable functions on the interval

It is proved that the space of functions constant on vertical lines is not complemented in the space of all measurable functions on the unit square (with the topology of convergence in measure). The analogous results is proved for the space of all measurable functions on the product of two probability spaces, one of which is atomless. This research was partially supported by National Science Foundation grant N.S.F. GP 28577.