# Direct construction of Lebesgue measure

I have seen two books for measure theory, viz, Rudin's, and Lieb and Loss, "Analysis".

Both use some kind of Riesz representation theorem machinery to construct Lebesgue measure.

Is there a more "direct" construction, and if so, what is a source?

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Royden's book on Real Analysis has the other "popular" approach. –  Nonliapunov Sep 25 '12 at 11:36
I'd also recommend Robert G. Bartle's "The elements of integration and Lebesgue measure" –  user8126 Sep 25 '12 at 11:55
I'd recommend, if you plan on staying as a member, to register your account. That way, you won't have login issues. –  mixedmath Sep 26 '12 at 2:23

The answer is, of course, yes. The other usual construction of the Lebesgue measure starts with the concept of outer measure. Then you introduce Caratheodory's definition of measurable set, and you develop the whole theory. This approach is presented in a clean way in Royden's book, Real analysis.

While the approach based on Riesz' representation theorem is good for the purposes of functional analysis and PDE theory, the second approach is useful for geometric measure theory, also called "italian measure theory". Moreover, Rudin's definition is rather abstract in nature, and students usually do not understand that the underlying idea is that of covering sets with intervals and summing up their lengths.

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