Convergence in Measure and in Category

Abstract

Abstract 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].