On the almost everywhere convergence of negative order Cesaro means of Fourier–Walsh series

In the paper is presented existence of an increasing sequence of natural numbers $M_{\nu} (\nu=0,1,...)$, such that for any $\varepsilon>0$ there exists a measurable set $E$ with a measure $\mu E > 1-\varepsilon>0$ such that for any function $f(x)\in L^1[0, 1]$ one can find a function $g(x)\in L^1[0, 1]$ which coincides with the function $f$ on $E$, and for any a $\alpha\neq 1, 2,...$ the Cesaro means $\sigma^{\alpha}_{M_{\nu}} (x,\tilde{f})\ (\nu=0,1,...)$ converges to $g(x)$ almost everywhere on $[0,1]$.