On convergence of Fourier series in complete orthonormal systems in the $L^1$-metric and almost everywhere

It is proved that if $\{\varphi_n(x)\}$ is a complete orthonormal system of bounded functions and $\varepsilon>0$, then there exists a measurable set $E\subset[0,1]$ with $|E|>1-\varepsilon$ such that
1) for any function $f(x)\in L[0,1]$ there exists a function $g(x)\in L^1[0,1]$ with $g(x)=f(x)$ on $E$ and such that the Fourier series of $g(x)$ in the system $\{\varphi_n(x)\}$ converges in the $L^1$-metric; and
2) there exists a subsequence of natural numbers $m_k\nearrow\infty$ such that for any function $f(x)\in L^1[0,1]$ there exists a function $g(x)\in L^1[0,1]$ such that $g(x)=f(x)$ for $x\in E$, $\displaystyle\lim_{k\to\infty}\sum\limits_{n=1}^{m_k}\alpha_n(g)\varphi_n(x)=g(x)$ almost everywhere on $[0,1]$, and $\{\alpha_n(g)\}\in l_p$ for all $p>2$, where $\displaystyle\alpha_n(g)=\int_0^1g(x)\varphi_n(x)\,dx$, $n=1,2\dots$ .