supremum as a norm What is it the intuition behind letting the limit of the $p$-norm $(\int |f(x)|^{p}dx)^\frac{1}{p}$ to be defined as $\sup f$?
Is this similar to taking as particular Hölder means functions$$M_p(s,t)=\left(\frac{s^p+t^p}{2}\right)^{\frac{1}{p}}$$
the two functions $M_{-\infty}(s,p)=\min\{s,p\}$ and $M_{\infty}(s,p)=\max\{s,p\}$?
 A: Here's how I think about it. The higher the value of $p$, the more the large values of the function are accentuated: where the function is approximately 1, the power of $p$ doesn't matter much; where it is less than 1, higher $p$-values decrease the contribution to the integral. Thus as $p \to \infty$, more and more contribution to the integral comes from the small regions where the function takes on its highest values.
Thus we can intuitively approximate $f$ by $\sup f$ on the set $A$ where $f$ is close to its maximum and $0$ everywhere else, giving $ \Vert f \Vert_p \approx \sup f \cdot\mu (A)^{1/p} \to \sup f$.
This is very much related to the Hölder means you mentioned - up to the factor of $2^{-1/p}$, the Hölder $p$-mean of ${s,t}$ is just the $L^p$-norm of the function $0 \mapsto s, 1\mapsto t$ on the measure space with two points of even measure; and this corrective factor vanishes as $p\to \infty$.
A: Assume that $f$ is a simple function, that is, $f=\sum_{k=1}^Nc_k\chi_{A_k}$, where $c_k$ are real numbers and $A_k$ pairwise disjoint measurable sets. Then 
$$\lVert f\rVert_{L^p}=\left(\sum_{k=1}^N|c_k|^p\mu(A_k)\right)^{1/p},$$
which is an extension of the definition of $M_p$ (with several entries, and this can be arranged into a convex combination, but not necessarily with the same coefficient). So the Hölder mean functions describe the behaviour of $L^p$ norms for simple functions, and of all bounded measurable functions on a finite measure space. 
