On the absolute convergence of Fourier–Haar series in the metric of $L^p(0,1)$, $0<p<1$

It is proved that for any $0<\epsilon<1$ there exists a measurable set $E\subset[0,1]$ with $|E|>1-\epsilon$ such that for any function $f(x)\in L^1[0,1]$ one can find a function $g(x)\in L^1[0,1]$ equal to $f(x)$ on $E$ such that its Fourier–Haar series converges absolutely in the metric of $L^p(0,1)$, $0<p<1$.