I have been studying measure on my own using Folland. I can generally follow the math, but I have lost sight of the big picture, at least on the real line. Basically, I am asking myself how Lebesgue measure on the real line is different from our everyday sense of length. What are we getting with Lebesgue measure that we cannot get from the everyday idea of length?
Provided of course that simple length has the property that the length of a countable union of disjoint sets is the sum of the individual lengths, what are the differences between simple length and Lebesgue measure?
Specifically, what are some sets we can measure with Lebesgue measure that we cannot with simple length? What are some sets that Lebesgue measures correctly, that everyday length does not?
Informally speaking, is it best to think of Lebesgue measure as a formalization of everyday length, a generalization of everyday length, or as a “correction” of everyday length?
I would certainly appreciate any comments and insights.
Note: We don’t need Lebesgue measure to calculate the measure of the Cantor set. We just add up the 1/3 parts we take out, which sum to 1, so the parts left (which are the Cantor set) have to measure zero. We can do all that with just simple length, and the every day idea that the length of a set must equal the sum of its disjoint parts.
Similarly, we don’t need Lebesgue measure to calculate the measure of the rationals since that measure is just the countable sum of points of zero measure.