Definition:Lebesgue Sigma-Algebra

The Lebesgue sigma-algebra on $$\R^N$$ is the smallest sigma-algebra over $$\R^N$$ which contains the Borel sigma algebra of $$\R^N$$, and that includes all null sets of $$\R^N$$.

This definition makes sense because the smallest sigma-algebra containing a given collection of sets is well-defined.