ABSTRACT
In this chapter, the author tells from context what σ-algebra of subsets are working within a given study area or problem. He adds more detail to the proofs of propositions to help readers build their analysis skills by looking hard at many interesting and varied proof techniques. If the author draws all the possible constant x and constant y lines corresponding to the endpoints, the author subdivides the original four intervals into many smaller intervals. The author does this by showing that Lebesgue outer measure is a metric outer measure. The translation invariance of Lebesgue measure is important in the construction of a non-Lebesgue measurable set. The author establishes this via two preliminary results which we put into Lemmas and then the final theorem and does many things with measurable sets. This material is a connection of sorts to the standard discussions of linear analysis.
