$A$-integrable martingale sequences and Walsh series

A sufficient condition for a Walsh series converging to an $A$-integrable function $f$ to be the $A$-Fourier's series of $f$ is stated in terms of uniform $A$-integrability of a martingale subsequence of partial sums of the Walsh series. Moreover, the existence is proved of a Walsh series that converges almost everywhere to an $A$-integrable function and is not the $A$-Fourier series of its sum.