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I was wondering if there is a set of algebraic "rules" for finding primitives of Lebesgue integrals as there is one for finding primitives of Riemann integrals. I.e. for $x^{n}$ the primitive is $\frac{x^{n+1}}{n+1} + C_{0}$ in the Riemann world first presented in school. Are there similar rules for working with Lebesgue integrable functions that would allow to compute the primitives of more functions (that is, functions that would not be Riemann integrable)?

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$\int D(x)\,dx=C$, where $D=\chi_{\mathbb Q}$ is the Dirichlet function. Does this count? – user31373 May 24 '12 at 0:38
Isn't $C$ equal to 0 in this case? Is $D$ the only function for which we have an "easy" primitive? – Frank May 24 '12 at 0:46
All of the functions I can think of with a reasonably nice antiderivative are continuous, so Riemann integrable (on intervals). What kind of functions do you have in mind? – Qiaochu Yuan May 24 '12 at 0:47
No, I think $C$ can be any real value (but now we are getting into the question of what $\int f(x)\,dx$ means, which is controversial). More seriously, the issue with your question is that you are asking for something "algebraic" (whatever that means) but not Riemann integrable. For any reasonable value of "algebraic" (maybe "elementary" is a better word), such functions are piecewise continuous, as Qiaochu already said. – user31373 May 24 '12 at 0:58
I do not have any specific function in mind. I was just comparing the way Riemann integration is introduced in class then leads to rules such as the one I mention above and the corresponding pages of exercises (and even MIT competitions), when nothing similar happens in the class on Lebesgue integration, as far as I can tell. – Frank May 24 '12 at 2:05

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