When Stein and Shakarchi (2009) navigate the properties of Lebesgue measurable sets, they provide the following property
If $m_*(E)=0$, then $E$ is measurable. In particular, if $F$ is a subset of a set of exterior measure $0$. then $F$ is measurable.
My Question:
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Any negligible volume set is measurable. Is this important? If so, in what way? Why do we care?
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When we look at $F$, is this a subset of a negligible volume set? So like something inside a point? I am trying to visualize. Why do we care if $F$ is measurable or not? How do we interpret a subset of a negligible volume set?
The authors give an example of the Cantor set being measurable with measure 0.
Noate Bene: I used the term "negligible volume set" loosely. By this I meant, if you try to measure something from outside with an open set, the "mass" is 0. An intuitive example that I can think of is a point? What are some other examples?
Reference:
Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.
Best Answer
A measure is a way of consistently assigning a size to sets. The more sets you can assign a measure to, the better - but you need to be careful that you don't run into paradoxes. That's the reason that the formalism of measure theory was introduced - because there used to be too many paradoxes in analysis before the introduction of this formalism. There's nothing stopping me from defining a wacky measure where some huge sets are assigned no measure and some other relatively small sets are assigned a lot of measure. For example, I can consider counting measure on the rationals: this assigns measure zero to all irrational numbers (an uncountable set) but infinite measure to the rationals (a much smaller countable set). So measure zero in this case does not mean negligible by any normal definition.
When some describes a measure, it comes in two parts: which sets are going to be assigned a measure (called "the $\sigma$-algebra") and what values will they be assigned. There are some situations where we can automatically extend the $\sigma$-algebra (and the measure) in a consistent way. This is known as "completing a measure": for example, there is unambiguously only one "reasonable" measure that can be assigned to any subset of a measure zero set: and it is zero. You may have encountered this in the most common situation, where we complete the Borel $\sigma$-algebra into the Lebesgue $\sigma$-algebra. It makes the measure more powerful, since there are sets whose measure was not previously defined but which now have a well-defined measure after the extension.
Too long; didn't read summary: It boils down to (somewhat pedantic) definitions, we care only if we are being very precise about the collection of sets our definitions apply to.
By the way, measure theory may turn a lot of things on their head but not the point: $\bullet$
A point is a point and it does not sprout any new subsets just because we start considering measures. Others already gave good explanations in the comments about this, but to summarize the sentiment: a point is not a general enough example of what a negligible set can look like. Think of a complicated but sparse set - exactly like the Cantor dust - to get a feeling for what a more typical example of a Lebesgue-negligible set looks like. (There are more exotic examples, like the random set of points at which Brownian motion equals zero for example.)