Suppose you have a random variable $X_1:\Omega \to \mathbb{R}$, which is $\mathcal{F}$ measurable and a random variable $X_2:\Omega \to \mathbb{R}$, which is $\mathcal{F}_t$ measurable with $\mathcal{F}_t\subset \mathcal{F}$, where $(\mathcal{F}_t)$ is a filtration with the usual condition and $\mathcal{F}_t\subset \mathcal{F}$ for all $t$. Now suppose we have $X_1=X_2$ a.s. Is $X_1$ then also $\mathcal{F}_t$ measurable? I think it is true, using that $\mathcal{F}_0$ contains all the null set of $\mathcal{F}$, but I'm not able to wirte down a rigorous proof. If someone could explain / prove why this holds (if it is indeed true), that would be very helpful.
hulik