Majorants and uniqueness of series in the Franklin system

It is proved that if a series in the Franklin system converges almost everywhere to a function $f(t)$ and the distribution function of the majorant of partial sums satisfies the condition
$$ \operatorname{mes}\bigl\{t\in[0,1]:s(t)>\lambda\bigr\} =o\biggl(\frac 1\lambda\biggr) $$
as $\lambda\to\infty$, then this series is a Fourier series for Lebesgue integrable functions $f(t)$. In the general case the coefficients of the series are reconstructed by means of an $A$-integral.