Convergence in measure of logarithmic means of quadratical partial sums of double Walsh–Fourier series

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh–Fourier series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of LlnL(I2), the set of the functions having logarithmic means of quadratical partial sums of the double Walsh–Fourier series convergent in measure is of first Baire category.