I read that the Minkowski integral inequality, which I knew for Riemann integrals on $[a,b]$, holds for Lebesgue integrals in the following form:$$\forall p\geq 1\quad\quad\Bigg(\int_X |f+g|^p d\mu\Bigg)^{\frac{1}{p}}\leq \Bigg(\int_X |f|^p d\mu\Bigg)^{\frac{1}{p}}+\Bigg(\int_X |g|^p d\mu\Bigg)^{\frac{1}{p}}$$I have been able to prove it to myself except for a thing: how do we know that $|f+g|^p$ is summable if $|f|^p$ and $|g|^p$ are? Can anybody give a proof of that? I thank you all!!!
EDIT: removed sentence where I mistakenly wrote that if $|f|$ and $|g|$ are measurable then $|f+g|$ is.
$$|f+g|^p \leq (|f| + |g|)^p \leq \left(2\max\{|f|,|g|\}\right)^p \leq 2^p \max\{|f|^p,|g|^p\} \leq 2^p(|f|^p+|g|^p)$$