I am interested in Lebesgue Measure and Integration.
so i have 2 questions. i want to learn that;
1) Can you explain me a set that is measurable but is not Lebesgue Measurable?
2) How may i study Lebesgue Integration without Lebesgue Measure Space?
I read an article about it. Here is a referrence.
https://www.jstor.org/stable/2324331?origin=crossref&seq=1 you may look it.
another referrence is also https://www.quora.com/How-can-I-study-the-measure-theory-without-studying-Lebesgues-measure
May you give some details ?thanks.
One route to Lebesgue integration without constructing the Lebesgue measure first is the Riez method where we first define the integral for step functions and then pass to functions that are the pointwise limit of step functions almost everywhere. The measure is then recovered by integrating the indicator function on a given set. A great book that covers this is Lebesgue Integration by Soo B. Chae.
Another route is the Lax method where we first consider the completion of the $L^{1}$ space prior to either the measure or the integral. This is covered in Functional Analysis by Peter Lax as well as in A Comprehensive Course in Analysis by Barry Simon.
As for your first question, the standard example of a set that is not Lebesgue measurable is the Vitali set.