On the $L^p_\mu$-strong property of orthonormal systems

Let $\{\varphi_n(x)\}$ be a system of bounded functions complete and orthonormal in $L^2_{[0,1]}$ and assume that $\|\varphi_n\|_{p_0}\leqslant\mathrm{const}$, $n\geqslant 1$, for some $p_0>2$. Then the elements of the system can be rearranged so that the resulting system has the $L^p_\mu$-strong property: for each $\varepsilon>0$ there exists a (measurable) subset $E\subset[0,1]$ of measure $|E|>1-\varepsilon$ and a measurable function $\mu(x)$, $0<\mu(x)\leqslant 1$, $\mu(x)=1$ on $E$ such that for all $p>2$ and $f(x)\in L^p_\mu[0,1]$ one can find a function $g(x)\in L^1_{[0,1]}$ coinciding with $f(x)$ on $E$ such that its Fourier series in the system $\{\varphi_{\sigma(k)}(x)\}$ converges to $g(x)$ in the $L^p_\mu[0,1]$-norm and the sequence of Fourier coefficients of this function belongs to all spaces $l^q$, $q>2$.