Non-measurable set

Remarks

Example: on $X = \set{1, 2}$ with $\Sigma = \set{\varnothing, X}$, the set $\set{1}$ is non-measurable. Example: a Vitali set in $[0, 1]$ is not Lebesgue measurable. Non-example: every interval, every open set and every closed subset of $\R^n$ is Lebesgue measurable, so none of them is non-measurable. For Lebesgue measure, $A \subseteq \R^n$ is non-measurable if and only if there is a set $B$ with $\lambda^*(B) < \lambda^*(B \cap A) + \lambda^*(B \setminus A)$, where $\lambda^*$ is Lebesgue outer measure. Every subset of $\R^n$ with positive outer measure contains a non-measurable subset. Being non-measurable depends on $\Sigma$: a set that is not a Borel set may still be Lebesgue measurable.