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Integral Measure and Derivative: A Unified Approach
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The book is quite good, I'd say; and yes, the Daniell approach is equivalent. You won't lose your time trying to nibble a bit at it.

Perhaps it's a bit dry by contemporary standards, try to find something more "modern"?

(That being said there is much drier, e. g. Saks' *Theory of the Integral* with terrible typography, each page looks like a solid, impenetrable wall.)
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Personally i love axlers book on the topic. Its free too so thsts helpful
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Soft & gentle:

Capinski & Kopp: “Measure Integral and Probability”
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For a pretty vanilla introduction to measure theory and integration you can try the first say 11 chapters of Bass’s “Real Analysis for Graduate Students”.

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