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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.2012-09-25
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    I'd also recommend Robert G. Bartle's "The elements of integration and Lebesgue measure"2012-09-25
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    I'd recommend, if you plan on staying as a member, to register your account. That way, you won't have login issues.2012-09-26

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