W. Orlicz in 1951 has observed that if {fn( ·, y)}n∈N converges in measure to f( ·, y) for each y∈[0,1], then {fn}n∈N converges in measure to f on [0,1]×[0,1]. The situation is different for the convergence in category even if we assume the convergence in category of sequences {fn( ·, y)}n∈N for each y∈[0,1] and {fn(x, · )}n∈N for each x∈[0,1].
English