Does $X\in L^p(\mathbb{P})$ imply $Y\in L^p(\mathbb{P})$

Question

Let $X$ and $Y$ be two i.i.d. real random variables, and suppose $X\in L^p(\mathbb{P})$. Does this imply that $Y\in L^p(\mathbb{P})$? And if so, how does one prove this?

Answer 1

Since $X$ and $Y$ have the same law, it follows that $$ \mathbb{E}[|X|^p]=\int_{\mathbb{R}}|x|^p\;d\nu_X(x)=\int_{\mathbb{R}}|x|^p\;d\nu_Y(x)=\mathbb{E}[|Y|^p] $$

In particular, $\mathbb{E}[|X|^p]$ is finite if and only if $\mathbb{E}[|Y|^p]$ is.