Lebesgue measurability is equivalent to open, closed, $F_\sigma$, or $G_\delta$ approximation up to null sets.Suggest a slogan
Let \(m_*\) be the outer measure on \(\RR^n\) and let \(E\subseteq \RR^n\). The following are equivalent:
\(E\) is Lebesgue measurable: for every \(\varepsilon>0\) there exists an open set \(G\supseteq E\) with \(m_*(G\setminus E)<\varepsilon\).
For every \(\varepsilon>0\) there exists a closed set \(F\subseteq E\) with \(m_*(E\setminus F) < \varepsilon\).
\(E = H \cup Z\) with \(H\) an $F_\sigma$ set and \(Z\) a null set.
\(E = V\setminus Z\) with \(V\) a $G_\delta$ set and \(Z\) a null set.
If \(m_*(E)<\infty\), these are also equivalent to:
- For every \(\varepsilon>0\) there exists a compact set \(K\subseteq E\) with \(m_*(E\setminus K) < \varepsilon\).
The finiteness hypothesis in (5) cannot be dropped: \(E=\RR^n\) is measurable, but \(m_*(\RR^n\setminus K)=\infty\) for every compact \(K\subseteq\RR^n\).
What conditions each characterise measurability of a set $E \subset \RR^n$?