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Is Lebesgue's integral the only definition of integral for which the $L^1$ space is a complete vector space (for a natural norm on it)?

I'm mostly interested on integrals of functions $f:\Bbb R^n \to \Bbb R$, and where the integral is an explicit operator (it must have a "practical utility", and we should be able to compute some integrals with it)

An integral $I$ must verify :

  • $I(\lambda f+g) = \lambda I(f)+I(g)$ (linear)
  • $I(f)\leq I(g)$ if $f \leq g$
  • $I(f) = \int_{\Bbb R^n} f(x) dx$ for every continuous function (and $\int$ is the Riemann integral)

Thanks

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  • $\begingroup$ What counts as a "definition of integral"? What sort of domain are you considering? $\endgroup$ – Eric Wofsey Dec 4 '15 at 16:33
  • $\begingroup$ @EricWofsey : added some precisions $\endgroup$ – Tryss Dec 4 '15 at 16:53

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