Abstract
It is shown that there exists a function $U\in L^1([0,1)^2)$ such that for each $\varepsilon \gt 0$ one can find a measurable set $E_\varepsilon \subset [0,1)^2$ with $|E_\varepsilon | \gt 1-\varepsilon $ such that $U$ is universal for the space $L^{1}(E_
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