In my measure theory course we studied norms $L_p$ and no other norms. For proofs we used exclusively the trick known as Hoelder's inequality which works only on $L_p$ norms. I disliked it very much so I found a proof for Minkowski's integral inequality which (slightly longer but) works for all norms that are monotone in this sense:
- $f\le g\;\;\;\text{a.e.}\;\Rightarrow\;|f|\le|g|\;\;\;\;\;$ (also, $\;\;f=g\;\;\;\text{a.e.}\;\Rightarrow\;|f|=|g|$),
- $\{f_n\}_{n\in\mathbb{N}}\;\;\text{is non-decreasing w.r.t. the relation above}\;\Rightarrow\;\lim|f_n|=|\lim f_n|$,
- $\{f_n\}_{n\in\mathbb{N}}\;\;\text{is non-increasing w.r.t. the relation above}\;\;\land\;\;|f_1|<\infty\;\Rightarrow\;\lim|f_n|=|\lim f_n|$.
Unfortunately, I couldn't prove it for all norms, but this was as general as I could imagine a norm be. Then, the total variation norm was substantially different, so I proved that case too:
Let $f:[0,1]^2\rightarrow\mathbb{R}$ be a Lipschitz continuous function (it also holds for more general functions, but let's keep it simple). Let $H:\mathbb{R}^{[0,1]}\rightarrow\mathbb{R}$ be the norm that maps (Lipschitz continuous) functions $f\in\mathbb{R}^{[0,1]}$ into $|f(0)|+\text{Var}_f[0,1]$. Let $f_x:[0,1]\rightarrow\mathbb{R}:y\mapsto f(x,y)$ and let $f^y:[0,1]\rightarrow\mathbb{R}:x\mapsto f(x,y)$ and let $\lambda$ be the Lebesgue measure. Then we have: $$H\bigg(x\mapsto\int f_xd\lambda\bigg)\le\int\Big(y\mapsto H(f^y)\Big)d\lambda.$$
On the other hand, I could not find any norm for which this inequality doesn't hold.
The question:
Is there any simple norm for which the Minkowski's integral inequality doesn't hold? What is the more precise relationship with the integral operator that the norm has to have in order to satisfy this inequality?
(The "definition" of the Minkowski's integral "inequality" in case of the general norm is like the one I gave for the total variation norm.)