Minkowski's Inequality/Lebesgue Spaces

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $f, g: X \to \R$ be $p$-integrable, that is, elements of Lebesgue $p$-space $\map {\mathcal L^p} \mu$.

Then their pointwise sum $f + g: X \to \R$ is also $p$-integrable, and:


 * $\norm {f + g}_p \le \norm f_p + \norm g_p$

where $\norm {\, \cdot \, }_p$ denotes the $p$-seminorm.