I would like to see the historical references for the following sequence of events:
1) When outer measure defined first time?
2) When it is proved that the outer measure is not countable additive?
3) When it was realized that it is not possible to define countably additive measure on all subsets of Euclidean space (which generalizes usual length, area, volume etc.?)
4) When Lebesgue measure is defined? (Possibly, it would be by Lebesgue; but I am not aware of the history of this subject.)
(Is there translation of Lebesgue's original work on measure theory and integration?)
The theory of outer measures was basically developed by Carathéodory. A reference would be C. Carathéodory, Vorlesungen über reelle Funktionen, 1st ed, Berlin: Leipzig 1918, 2nd ed, New York: Chelsea 1948.
Vitali found the first example of a non-measurable set in 1905.