Almost Sure Equality Preserves Square-Integrability and the Mean-Square Norm

Let $(\Omega,\mathcal{F},P)$ be a probability space, and let $X$ and $Y$ be random variables on it that are almost surely equal, in the sense that there is an event $A'\in\mathcal{F}$ with $P(A')=0$ such that $X(\omega)=Y(\omega)$ for every $\omega\in\Omega\setminus A'$.

If $X$ is square-integrable, then $Y$ is square-integrable, the expectations $\mathbb{E}[X^{2}]$ and $\mathbb{E}[Y^{2}]$ are equal, and $X$ and $Y$ have the same mean-square norm.