I have this following task for Lebesgue spaces $\mathscr{L}^{p}$ with $p<1$:
For $p\in(0,1)$ give an example of a measure space $(X,\mathscr{A},\mu)$ and $f,g\in\mathscr{L}^{p}(X,\mathscr{A},\mu;\mathbb{C})$ with $$ \|f+g\|_{p}>\|f\|_{p}+\|g\|_{p}. $$ I now that this is basically the opposite of the Minkowski inequality, but I do not really know what to do here. I do not need to prove something here but I need an example, as the task states. Does anybody know what would be a good example here?
Based on the discussion under my question I have now a complete solution to the task:
Let $X=\mathbb{R}$ and $\mathscr{A}=\mathscr{B}(X)=\mathscr{B}(\mathbb{R})$ and the Lebesgue measure $\lambda^{1}$. Let $f(x)=\chi_{[1,2]}(x)$ and $g(x)=\chi_{[3,4]}(x)$. For the norm of $f$ and $g$ we then have:
$$||f||_p=||g||_p=\left(\int_X|f|^pd\lambda^1\right)^{1/p}=(\lambda^1([1,2]))^{1/p}=1$$
For $f+g$ we then have:
$$||f+g||_p=\left(\int_X|f+g|^pd\lambda^1\right)^{1/p}=(\lambda^1([1,2]\cup[3,4]))^{1/p}=(\lambda^1([1,2])+\lambda^1([3,4]))^{1/p}=2^{1/p}$$
In total we have:
$$||f+g||_p=2^{1/p}>2=||f||_p+||g||_p$$ since $p\in(0,1)$.