Proof for a Markov process example (using measure theory)

Question

Consider the probability space $(\mathbb{R},B(\mathbb{R}),\delta_x)$ for a given $x\in\mathbb{R}$ (where $\delta_x$ is the Dirac measure) and define the process $X_t(\omega)=\omega - t$, for $t\geq 0$.

How does one prove that $X_t$ is a Markov process with respect to the filtration $\mathcal{F}_t = B(\mathbb{R})$?