Diffrent definition of Lebesgue integral Have you ever seen this?

In summary, the book 'ANALYSIS by Lieb and Loss' defines the Lebesgue integral in terms of Riemann integration, using the formula \int f d\mu : = \int_0^{\infty} \mu (\{x \in X : f(x) > t \}) dt where \mu is a measure and f is a measurable and non-negative function. This definition is not commonly seen in other books, but Hewitt and Stromberg's 'Real and Abstract Algebra' also provide a similar definition. For a better understanding, the link provided in the conversation can be helpful.
  • #1
lysangc
1
0
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 !

[tex]\int f d\mu : = \int_0^{\infty} \mu (\{x \in X : f(x) > t \}) dt[/tex]
of course, [tex]\mu[/tex] 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!

-------------------------------
P.S. Sorry for poor english..
 
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  • #2
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/
 

1. What is the Lebesgue integral and how is it different from other definitions of integration?

The Lebesgue integral is a mathematical concept used to calculate the area under a curve. It differs from other definitions of integration, such as the Riemann integral, in that it allows for integration over more complicated sets, such as unbounded or discontinuous functions.

2. How was the Lebesgue integral developed and who is credited with its creation?

The Lebesgue integral was developed by French mathematician Henri Lebesgue in the early 20th century. He built upon the work of other mathematicians, including Riemann and Borel, to create a more general and powerful definition of integration.

3. Can you provide an example of a function that can only be integrated using the Lebesgue integral?

Yes, a function that is discontinuous, such as the Dirichlet function, cannot be integrated using the Riemann integral, but can be integrated using the Lebesgue integral. This function is defined as 1 for rational numbers and 0 for irrational numbers.

4. What are the advantages of using the Lebesgue integral over other definitions of integration?

The Lebesgue integral has several advantages, including its ability to integrate over a wider range of functions, its ability to handle more complicated sets, and its ability to handle functions that are not necessarily continuous or differentiable.

5. Are there any limitations or drawbacks to using the Lebesgue integral?

One limitation of the Lebesgue integral is that it can be more difficult to compute in practice compared to the Riemann integral. It also requires a solid understanding of measure theory, which can be a barrier for some individuals. Additionally, for certain functions, the Lebesgue integral may not provide a more accurate result than the Riemann integral.

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