# 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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The most popular way is constructing it using the Caratheodory extension theorem, from Lebesgue outer measure. This approach is not very intuitive, but is a very powerful and general way for constructing measures.

An even more direct construction and essentially the one developed by Lebesgue himself defines Lebesgue measurable sets to be the ones that can be well approximated (in terms of outer measure) by open sets from the outside and by closed sets from the inside and shows that Lebesgue outer measure applied to these sets is an actual measure. You find this approach in A Radical Approach to Lebesgue's Theory of Integration by Bressoud and Real Analysis: Measure Theory, Integration, and Hilbert Spaces by Stein and Shakarchi.

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I actually find this reversed. Indeed, having seen the construction starting from outer measures first, the approach via RRT seems like bad pedagogy. In a functional analysis class last semester the lecturer didn't want to spend too much time on it so used the RRT approach quickly, and many who hadn't seen the other approach before struggled to gain intuition for actual construction and thought of it as a black box which gave the "intuitive" notion of measure. – Ragib Zaman Sep 25 '12 at 11:45
@RagibZaman The most intuitive approach to constructing Lebesgue measure is based on completing a certain pseudo-metric space defined in terms of outer measure. – Michael Greinecker Sep 25 '12 at 11:49