I am revising the lecture notes and i stumbled across this proposition:
Given $(\Omega, \mathcal{A}, P)$ a probability space, the following holds:
If a process $(X_t)_{t\geq 0}$ has continuous trajectories (that is the map $t\to X_t$ is continuous), then it is measurable (that is $X\colon(\Omega\times[0,\infty),\mathcal{A}\otimes\mathcal{B}([0,\infty)))\to(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ is measurable),
If $(X_t)_{t\geq 0}$ is also adapted to a filtration $(\mathcal{F}_t)_{t\geq 0}$ then it is progressively measurable w.r.t. $(\mathcal{F}_t)_{t\geq 0}$
In class, we proved the second claim, using the following method: fixed $T>0$, we defined $$X^n_t=\sum_{k=1}^{2^n}X_{\frac{k}{2^n}T}\, 1_{\left[\frac{k-1}{2^n}T,\frac{k}{2^n}T\right)}(t)+X_T\,1_{\{T\}}(t)$$ Since it is all $\mathcal{F}_T$-measurable, then we have that $X^n$ is measurable (as a process). Then, thanks to the continuity of the trajectories, we have that $$\forall t\in [0,T], \forall\omega\in\Omega, \quad X_t^n(\omega)\rightarrow X_t(\omega)\quad\text{as }n\rightarrow +\infty$$ and this basically gives us the thesis.
Now our professor said that the first claim is to be demonstrated in a similar way, but i don't really know how to start. Can someone answer me or give me an hint?