Strange definition of Lebesgue Integral !!!

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 !!!

$\displaystyle \int f d\mu : = \int_0^{\infty} \mu (\{x \in X : f(x) > t \}) dt$

of course, $\displaystyle \mu$ is measure, f is measurable.

LHS -> general (lebesgue) integration

RHS -> 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.. and I am not asking why equality holds..