Let \(m\) be Lebesgue measure on \(\RR^n\). A set \(A\subseteq\RR^n\) is null if for every \(\varepsilon>0\) there exist open sets \(U_1,U_2,\ldots\subseteq\RR^n\) with \(A\subseteq\bigcup_{j} U_j\) and \(\sum_j m(U_j) < \varepsilon\).
A set \(A\) is null if and only if its outer measure \(m_*(A)\) is \(0\). Indeed, \(m_*(A)\leq\sum_j m(U_j)\) for every such cover; conversely, if \(m_*(A)=0\), a cover of \(A\) by closed cubes of total volume less than \(\varepsilon/2^{n+1}\) can be replaced by the open cubes with the same centers and twice the side lengths, of total volume less than \(\varepsilon\).
What does it mean for a set to be null?