Diffrent definition of Lebesgue integral Have you ever seen this?

[lysangc]

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!

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P.S. Sorry for poor english..

 

[fresh_42]

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/