# Strange definition of Lebesgue Integral !!!

• Feb 16th 2009, 02:16 AM
ramsey88
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..
• Feb 16th 2009, 05:56 AM
CaptainBlack
Quote:

Originally Posted by ramsey88
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!!
-------------------------------
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
• Feb 16th 2009, 03:46 PM
ramsey88
Quote:

Originally Posted by CaptainBlack
I've seen something like this, I think it was in this

CB

No, the definition is nowhere in the book..

Somebody help me!
• Feb 16th 2009, 10:22 PM
CaptainBlack
Quote:

Originally Posted by ramsey88
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