Minkowski's Inequality/Lebesgue Spaces

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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

Let $f, g: X \to \R$ be $p$-integrable, i.e. elements of Lebesgue $p$-space $\mathcal{L}^p \left({\mu}\right)$.

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


 * $\left\Vert{f + g}\right\Vert_p \le \left\Vert{f}\right\Vert_p + \left\Vert{g}\right\Vert_p$

where $\left\Vert{\cdot}\right\Vert_p$ denotes the $p$-seminorm.