On the existence and structure of universal functions for weighted spaces $$L^1_\mu [0,1]$$

In the paper it is shown that there exist a set $$E\subset [0,1]$$ with measure being arbitrarily close to 1 and weighted function $$lt\mu(x)\le 1$$ $$\mu (x)=1$$ on E, so proper modification of an arbitrary $$f\in L^1[0,1]$$ outside E one can obtain $$\tilde{f}\in universal for space $$L^1_{\mu} respect Walsh system in sense signs.