$u$ in $L^p$ implies that $u+c$ is in $L^p$.

Question

Suppose that $\displaystyle\int_U|u(x)|^p\,dx<\infty$, where $U\subset\mathbb{R}^{n}$ is bounded and $p$ is a positive integer.

How can I prove that $\displaystyle\int_U|u(x)+c|^p\,dx<\infty$? where is a real contant.

Thanks!

Answer 1

$$\|u + c\|_p \le \|u\|_p + \|c\|_p.$$ Since you are on a set of finite measure, $\|c\|_p$ is finite.