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 $0\leq t_1<...<t_n\leq t$ and all $B_i\in \mathcal B(\mathbb R)$ ?

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


Yes, they are the same. Just verify that each is contained in the other.


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