= Null-set limsup cover by finite arc families
{title2=$E\subset\limsup_j F_j,\qquad\sum_{I\in F_j}|I|<\epsilon_j$}
For any circle null set and any positive sequence $\epsilon_j$, take infinitely many countable covers with summable total lengths. Regard their arcs as a two-index array. Choose increasing finite-square cutoffs so the length sum outside each square is smaller than the next desired bound. The finite square annuli then give finite arc families with the stated small lengths. Every point belongs to some arc in every row, hence to infinitely many annular families. This handles dense and nonclosed null sets without claiming that they have finite small covers.
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