Measure can really be built up -in steps-!!
(Instead of jumping to the end)

- Jordan measure/content
  use finitely many rectangles
  (open or closed or half-open);
  this suffices for all conventional sets.

Incidentally, if you use Jordan content and a Lebesgue-y definition of integral, then you get exactly the Riemann integral (indeed, this is what -Jordan- did, not Lebesgue. Lebesgue's insight was to improve the measuring of sets).

This is simply formalizing Eudoxus's method of exhaustion.

- Borel
  [I don't know if this came before or after Lebesgue]
  Here you allow -countably- many
  (and complements!)
  you then need to throw in these sets as "measurable"
  and take a closure.
  
NB: this does -not- arise under Jordan:
Anything that can be approximated by finitely many Jordan measurables
can be approximated by finitely many rectangles (finite & finite = duh).

In other words, the extension is:
  + allow (countable!) infinites
  + ...and take an (algebraic!) closure
  ...which yields a $\sigma$-algebra

- Lebesgue
  The added refinement is to priviledge null-sets
  (yielding a -complete- sigma-algebra),
  and toss 'em in too.

NB: the key difference is that Lebesgue (or Borel?)
allows infinite sets to -start-,
instead of starting with finite ones and taking a limit.
[Switching limits makes a difference, as usual.]
This makes a difference, as attested by the rationals.
I suspect that the issue is that for -closed- sets, Jordan is fine,
but if you want more disgusting sets, it's not enough.

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Note that a slick technical way of doing it (which offers some intuition
but is rather alien) is to say that a set is measurable if it can be
approximated from within and without by sets that differ by a null-set.

This reduces everything to defining null-sets,
and is conceptually nice is that it clarifies that it's this difference -set-
that is the issue,
but breaking this down into "measure inside; measure outside"
relates everything to basic measuring and integrals.
