Lebesgue integral

[Kount]

How can it be justified that the Lebesgue integral really gives something like the area under the curve of the function?

How do we justify countable additivity of the Lebesgue measure and what is the crucial point that we can say that the supremum of the integrals of the simple functions (smaller than the integrated non-negative function) is something like the area under the curve of the integrated function? Is it because there is a sequence of simple functions that converge uniformly against the function?

Thanks in advance!

 

[pka]

How can it be justified that the Lebesgue integral really gives something like the area under the curve of the function?
How do we justify countable additivity of the Lebesgue measure and what is the crucial point that we can say that the supremum of the integrals of the simple functions (smaller than the integrated non-negative function) is something like the area under the curve of the integrated function? Is it because there is a sequence of simple functions that converge uniformly against the function?

You are asking to be given most of the material covered in six weeks of a real variables course. We can't do that. But I will suggest that you look at one of the most readable text:
MEASURE and INTEGRATION by Sterling K. Berberian.

 

[Kount]

But at least you start with the definition of measure, so you have to claim countable additivity before you can do anything else.