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So far I've understood that the framework to defining Lebesgue integrals is totally different from Riemann integrals. But, how do the actual computations of integrals to differ?

In Riemann integrals, I understand computation of an integral as, algebraically manipulating the integral till its in form where the expression inside the integrand is the derivative of something we know. So, when we say the amount of Lebesgue integrable functions are more, what are the new computations allowed to calculate their value?

Clemens Bartholdy
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  • One of the things that is newly allowed is "changing the value of the function on a set of Lebesgue measure zero". – JonathanZ Apr 03 '23 at 21:36
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    In practice, the only way to evaluate a Lebesgue integral — unless you can apply some sort of dominated convergence limiting argument — is to evaluate a Riemann integral. – Ted Shifrin Apr 03 '23 at 21:43
  • What about $\int_{[0,1]} D(x)dx$, where $D=\chi_{\mathbb{R}-\mathbb{Q}}$? – Asigan Apr 05 '23 at 08:35

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