# What is exactly $\sigma (X_s\mid s\leq t)$ for $(X_t)$ a stochastic process?

Let $$(\Omega ,\mathcal F,\mathbb P)$$ a probability space. Let $$(X_t)$$ a stochastic process. I know that $$\mathcal F_t:=\sigma (X_s\mid s\leq t)$$ is the smallest $$\sigma -$$algebra s.t. $$X_s$$ is measurable for all $$s\leq t$$. I'm not sure what that exactly means. Does it means that :

1) $$\mathcal F_t$$ is the $$\sigma -$$ algebra generated by $$X_s^{-1}(B)\in \mathcal F_t$$ for all $$s\leq t$$ for all $$B\in \mathcal B(\mathbb R)$$ ?

or does it means that

2) $$\mathcal F_t$$ is the $$\sigma -$$algebra that is generated by $$X_{t_1}^{-1}(B_1)\cap ...\cap X_{t_n}^{-1}(B_n)$$ for all $$0\leq t_1<... and all $$B_i\in \mathcal B(\mathbb R)$$ ?

I have the impression that $$1)$$ and $$2)$$ are in fact equivalent, no ?