A Rienmann integral is defined as:

$\int_a^b f(x)\ dx=\displaystyle\lim_{n\rightarrow\infty}\displaystyle\frac{1}{n}\displaystyle\sum_{i=0}^{n}f\left(\displaystyle\frac{b-a}{n}\cdot i+a\right)$

I was wondering if their exists literature on a more generalized concept which is derived from the $p$-norm:

$\int_a^{b,p} f(x)\ dx=\sqrt[p]{\displaystyle\lim_{n\rightarrow\infty}\displaystyle\frac{1}{n}\displaystyle\sum_{i=0}^{n}\left|f\left(\displaystyle\frac{b-a}{n}\cdot i+a\right)\right|^p}$

The definition can both be seen as a generalized integral or generalized norm...

  • 1
    $\begingroup$ What about p-norms : $(\int |f(t)|^p)^{1/p}$ ? $\endgroup$
    – Bertrand R
    Nov 12, 2013 at 13:33
  • $\begingroup$ Is their some calculus defined on that part? $\endgroup$ Nov 12, 2013 at 13:35
  • $\begingroup$ You can show that for p norms when p goes to infinity you have on interval I for the sake of simplicity here : $(\int_I |f|^p)^{1/p}\rightarrow ||f||_\infty = \sup_I |f|$ (this comment was related to yours which you removed) $\endgroup$
    – Bertrand R
    Nov 12, 2013 at 13:37
  • $\begingroup$ Appologies for post-editing. Thanks $\endgroup$ Nov 12, 2013 at 13:45

1 Answer 1


What you are trying to define amounts exactly to the $p$-norm: $$ \sqrt[p]{\displaystyle\lim_{n\rightarrow\infty}\displaystyle\frac{1}{n}\displaystyle\sum_{i=0}^{n}\left|f\left(\displaystyle\frac{b-a}{n}\cdot i+a\right)\right|^p}=\left(\int_a^b|f(x)|^p\,dx\right)^{1/p} $$ whenever $|f|^p$ is integrable.


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