Minkowski Inequality states that: $$\left(\int (f+g)^p \right)^{1/p} \leq \left( \int f^p \right)^{1/p} + \left( \int g^p \right)^{1/p}.$$ (Assumption given is that $f, g$ are measurable on space $(X, \mathcal A, \mu))$.
Our class proof begins as:
$$\int (f+g)^p = \int f(f+g)^{p-1} + \int g(f+g)^{p-1}.$$
I am not sure this is a valid statement. This statement implicitly says that if $(a(x)+b(x))\psi$ is an integrable function, then $a\psi$ and $b\psi$ are both integrable. Why is it true in general? More explicitly, am I guaranteed that $\int |a\psi| < \infty$ and $\int |b\psi| < \infty$?
Best Answer
doesn't seem so. for example if $a(x)=-b(x)=c\neq 0$ for a.e. $x$ and $\psi$ not integrable but a.e. finite function, then $(a(x)+b(x))\psi$ is integrable but not $a\psi$ and $b\psi$.