Let $g(x)$ be a continuous function defined on the interval $[0, 1]$. Is it true that: $$\int_0^1\int_0^1|g(x)+g(y)| \, \mathrm{d}x \, \mathrm{d}y \geq \int_0^1|g(x)| \,\mathrm{d}x ?$$
What I’ve done: I’ve used Holder’s inequality and got $$\int_0^1\int_0^1|g(x)+g(y)| \, \mathrm{d}x \, \mathrm{d}y \geq \int_0^1\left|g(x)+ \int_0^1g(y) \, \mathrm{d}y\right| \, \mathrm{d}x. $$
I don’t know how to proceed.
