Given a probability space $(\Omega, \mathcal{F}, P)$ and a stochastic process $Y_t$ with continuous path, are there two ways of understanding $$ \sigma\{Y_s, 0\leq s \leq t\}?$$
First is looking at $\sigma\{Y_s, 0\leq s \leq t\}$ as a sub $\sigma$-algebra of $\mathcal{F}$, and it is generated by the family of r.v.s $Y_s: \Omega \rightarrow \mathbb{R}$ for $0\leq s\leq t$. We often call this the filtration $\mathcal{F}_t^Y$.
The second way is we look at the stochastic process as a random variable $$Y:\Omega \rightarrow C([0,t],\mathbb{R})$$ where the space of continuous functions $C$ has the sup-norm and Borel $\sigma$-algebra $\mathcal{B}(C)$. Now $\sigma\{Y_s, 0\leq s \leq t\}$ can be seen as a sub $\sigma$-algebra of $\mathcal{B}(C)$ which is generated from the single random variable $Y$.
Are these understandings correct?
Also we know $Y$ induces a probability measure on $\mathcal{B}(C)$, which gives us a probability space $(C, \mathcal{B}(C), \mu_Y) $. For a second stochastic process $X_u$ where $0\leq u \leq t$, are there any connections between the conditional expectation in the probability space $$E(X_u | \sigma\{Y_s, 0\leq s \leq t\})$$ and the conditional expectation in the function space $$E^*(X_\cdot | \sigma\{Y_\cdot\})?$$