Let $(X,\mathscr{M},\mu)$ be a measure space, and let $f,g:X\to[0,+\infty]$ be measurable functions. Let $1<p<\infty$, then $$\left( \int\limits (f+g)^{p} \, d\mu \right)^{1/p}\le\left( \int\limits f^{p} \, d\mu \right)^{1/p}+\left( \int\limits g^{p} \, d\mu \right)^{1/p}$$ or equivalently: Let $1\le p \le \infty$ and $f,g\in L^{p}(X,\mathscr{M},\mu)$, then $f+g\in L^{p}(X,\mathscr{M},\mu)$ and $$\|f+g\|_{p}\le\|f\|_{p}+\|g\|_{p}$$