Previously, we defined a measure m*(I) to be the infimum of the sum of the components (covers) of the measure (i.e. due to the measure being additive). We talked about moving forward and refining our example. In some texts, they continue with denoting a measure using m(I). However, most texts move forward with a different… Read More Lebesgue Theory Part 3

Part 1 of this post focused on an introduction into sets, what a ring is, what a sigma-ring is, what does it mean to be additive, independent, and countably additive. We finished with a theorem that addressed the essence of countable additivity; this theorem set up the needed tools to evaluate some important aspect of… Read More Lebesgue Theory Part 2

What is a set function? In fact, what is a function? Before jumping into the deep water, let us take a moment to review the idea of a function, what it is, and how it is useful to us. 1) Definition: Given two non-empty sets and a function  is a rule (i.e. a set of instructions) where each… Read More Lebesgue Theory Part 1 (Set Functions)