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Main Question or Discussion Point
Question 1: Can someone please explain to me on page 2 how is it that the inequality halfway through the page follows from the comparison property?
I don't see how. To me it just seems like an obvious fact since for each i, m(f)v(R_i) is less than or equal to the integral of f over R_i just by definition of the ordinary integral. I don't see how the comparison property comes into play here.
Question 2: The text says on page 3, "This corollary  corollary 15.5  tells us that any theorem we prove about extended integrals has implications for ordinary integrals. The change of variables theorem, which we prove in the next chapter, is an important example."
I don't see how. First of all, the extended integral is only defined for open sets. In this corollary the set S is completely arbitrary so it doesn't necessarily have to be open. Second, the function is bounded which in the case of an extended integral doesn't necessarily have to be so  it could be unbounded in which case the ordinary integral doesn't make sense. So how could this corollary show that ANY theorem we prove about extended integrals have implications for ordinary integrals?
Any help would be appreciated. Thanks!
I don't see how. To me it just seems like an obvious fact since for each i, m(f)v(R_i) is less than or equal to the integral of f over R_i just by definition of the ordinary integral. I don't see how the comparison property comes into play here.
Question 2: The text says on page 3, "This corollary  corollary 15.5  tells us that any theorem we prove about extended integrals has implications for ordinary integrals. The change of variables theorem, which we prove in the next chapter, is an important example."
I don't see how. First of all, the extended integral is only defined for open sets. In this corollary the set S is completely arbitrary so it doesn't necessarily have to be open. Second, the function is bounded which in the case of an extended integral doesn't necessarily have to be so  it could be unbounded in which case the ordinary integral doesn't make sense. So how could this corollary show that ANY theorem we prove about extended integrals have implications for ordinary integrals?
Any help would be appreciated. Thanks!
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