$|X|\in\mathcal{L}^p\Leftrightarrow X\in\mathcal{L}^p\hspace{0.3cm}\forall p>1$?

Question

Given a probability space $\left(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P}\right)$ and a random variable $X$ defined on it, I know that it holds true that $|X|\in\mathcal{L}^1 \Leftrightarrow X\in\mathcal{L}^1$.

Does it hold true that $|X|\in\mathcal{L}^p \Leftrightarrow X\in\mathcal{L}^p\hspace{0.3cm}\forall p>1$ as well?

Answer 1

We have $$|X| \in \mathcal{L}^p \iff \int ||X||^p < \infty \iff \int|X|^p < \infty \iff X \in \mathcal{L}^p$$

Answer 2

I understand that

$$\mathcal{L}^{p}=\{ X\colon \Omega\to \mathbb{R}\colon X\text{ is measurable and } E(|X|^{p})<\infty \}.$$

If $X\in \mathcal{L}^{p}$, we have by definition that $E(|X|^{p})=\int_{\Omega}|X|^{p}<\infty$, so $|X|\in \mathcal{L}^{p}$. Conversely, if $|X|\in \mathcal{L}^{p}$, then by definition $E(|X|^{p})<\infty$, so $X\in \mathcal{L}^{p}$.