On the limits of indetermination and on the set of limit functions of series in the Walsh system

The main result of the article is the following
Theorem. {\it If a series with respect to the Walsh system is summable $(C,1)$ on a set $E$ of positive measure to a finite function $f(t)$, the subsequence $\{S_{2^n}(t)\}$ of partial sums of this series converges almost everywhere on $E$ to the function $f(t)$.}
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