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In my analysis class we are seeing the so called Lebesgue-Radon-Nikodym Theorem. But we prove it the "old fashioned way" without using Hilbert space theory. More precisely, we prove the minimality property of the Jordan decomposition directly, and then use it to prove LRN theorem.

I am looking for textbook references that also have this approach. Do you know any?

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Folland's Real Analysis does it this way.

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