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Math Help - Strange definition of Lebesgue Integral !!!

  1. #1
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    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 !!!
    \int f d\mu : = \int_0^{\infty} \mu (\{x \in X : f(x) > t \}) dt
    of course, \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!!
    -------------------------------
    P.S. Sorry for poor english.. and I am not asking why equality holds..
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ramsey88 View Post
    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.
    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!!
    -------------------------------
    P.S. Sorry for poor english.. and I am not asking why equality holds..
    I've seen something like this, I think it was in this

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    I've seen something like this, I think it was in this

    CB
    No, the definition is nowhere in the book..

    Somebody help me!
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by ramsey88 View Post
    No, the definition is nowhere in the book..

    Somebody help me!
    IIRC the definition is not explicitly in the form you give, but it is equivalent, but then I might be mistaken about where I saw it.

    The book is not mine, but borrowing the copy of my colleage I find it is Appendix B, section 3, page 718. Which is different wording, but to a casual glance looks equivalent.

    CB
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