# Finding a counter example for Minkowski's integral inequality for $p=\infty$ [closed]

Dear All,

As we know that this following Miskowski's integral inequality is true for $1\leq p<\infty$

$[\int_{S_1}|\int_{S_2}F(x,y)d\mu_1(x)|^pd\mu_2(y)dy]^{1/p} \leq \int_{S_2}[\int_{S_1}|F(x,y)|^pd\mu_2(y)]^{1/p}d\mu_1(x)$

I think that Minkowski's integral inequality is not right for the case $p=\infty$. And I am trying to find a counter-example when $p=\infty$ but I have not had luck. Does anyone know any counter example for that case OR if you know how to prove that it is still right. Thank you very much and I really appreciate your help.

Phi

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