It is shown tlıat limit of the Walsh sumgn(x) S Xnk (x), 0 < x < l„ exists k—o and is integrable if and only if lim Xon < 00. Then gn converges in L l norm to g t n—>00 and for 0 < p < 1, Hm J ] g(x) - gn (x) |p dx = 0 .