Diffrent definition of Lebesgue integral Have you ever seen this?

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SUMMARY

The discussion centers on the definition of the Lebesgue integral as presented in the book "ANALYSIS" by Lieb and Loss, which contrasts with the traditional definition based on simple functions. Instead, it utilizes a formulation involving Riemann integration, specifically expressed as \(\int f d\mu = \int_0^{\infty} \mu (\{x \in X : f(x) > t \}) dt\), where \(\mu\) is a measure and \(f\) is a measurable, non-negative function. Participants seek references for similar definitions in other literature, noting that Hewitt and Stromberg's work provides an alternative approach to Lebesgue integration.

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  • Understanding of Lebesgue integration concepts
  • Familiarity with Riemann integration techniques
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  • Research the definition of Lebesgue integral in "Real and Abstract Analysis" by Hewitt and Stromberg
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lysangc
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I am studying 'ANALYSIS by Lieb and Loss '...

usually lebesgue integral is defined in terms of simple function

But

In this book, integral is defined in terms of Riemann Integration !

\int f d\mu : = \int_0^{\infty} \mu (\{x \in X : f(x) > t \}) dt
of course, \mu is measure, f is measurable, non-negative

LHS -> general (lebesgue) integration
RHS -> (improper) Riemann integration

Have you ever seen this definition in any other books?

If so, which book ? I need Reference .. HELP ME PLEASE!

-------------------------------
P.S. Sorry for poor english..
 
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Hewitt, Stromberg (Real and Abstract Algebra, GTM 25) define it as a sum over a dissection of ##X## instead of an integral. They give quite an elaborated introduction to Lebesgue integration. On a quick view I haven't found your exact definition, and I do not understand it, as it looks that it double counts a lot. Nevertheless, Hewitt, Stromberg are a good recommendation.

For a basic understanding see
https://www.physicsforums.com/insights/omissions-mathematics-education-gauge-integration/
 

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