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 !!!

of course, 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..