# Integral like a norm instead of sum

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...

• What about p-norms : $(\int |f(t)|^p)^{1/p}$ ? Nov 12, 2013 at 13:33
• Is their some calculus defined on that part? Nov 12, 2013 at 13:35
• 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) Nov 12, 2013 at 13:37
• Appologies for post-editing. Thanks Nov 12, 2013 at 13:45

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.