Representation of measurable functions almost everywhere by convergent series

In this paper it is proved that for a certain class of systems $\{\varphi _k\}$ (systems of type $(\mathrm{X})$) one may construct a series
\begin{equation} \sum^\infty_{k=1}a_k\varphi_k(t),\qquad t\in[0,1], \end{equation}
having the following properties:
1) $\lim_{k\to\infty}a_k\varphi_k(t)=0$ uniformly on the interval $[0,1]$.
2) For any measurable function $f(t)$ on the interval $[0,1]$ and for any number $N$, one can find a partial series
$$ \sum^\infty_{k=1}a_{n_k}\varphi_{n_k}(t),\qquad(N<n_1<n_2<\cdots), $$
from (1) which converges to $f(t)$ almost everywhere on the set where $f(t)$ is finite, and converges to $f(t)$ in measure on $[0,1]$.
3) If, in addition, the functions $\varphi_k$ ($k\geqslant1$) and $f$ are piecewise continuous and $\inf_{t\in[0,1]}f(t)>0$, then
$$ \sum^\infty_{k=1}a_{n_k}\varphi_{n_k}(t)\qquad\text{for all $t\in[0,1]$ and $m\geqslant1$}. $$
It is shown that systems of type $(\mathrm X)$ include, for example, trigonometric systems, the systems of Haar and Walsh, indexed in their original or a different order, any basis of the space $C(0,1)$, and others.
Bibliography: 19 titles.