Question:
In the typical construction of the Lebesgue Measure in 6 stages (eg. in Lebesgue Integration on Euclidean Space by Jones), authors often construct the measure on sets that have finite outer measure before sets with arbitrary measure.
In Jones, he writes,
We should like to say that a set is measurable if $\lambda^* (A) = \lambda_* (A)$ (We say that such sets exist in $\mathcal{L}_0$). But if $A$ is so large that $\lambda_* (A) = \infty$, then this definition would not restrict $A$ at all.
Why is this step necessary? In the final definition of the Lebesgue measure, we have $\lambda(B) = \sup \{ \lambda ( B \cap A ) | A \in \mathcal{L}_0 \}$ where $B$ is a set with arbitrary measure, not necessarily finite outer measure. If it's possible that $\lambda(B) = \infty$, then shouldn't the definition of measurable in the quotation above suffice?