Let $B$ be a standard Brownian motion on a probability Space $(\Omega, \mathcal{F}, P)$ and let $\mathbb F:=(\mathcal{F}_t)_{t\in [0,T]}$ denote the natural filtration, i.e. $\mathcal{F}_t = \sigma(W_s: \ 0\leq s\leq t) \vee\mathcal{N}$.
My question is simple: If $f(s)$ is a deterministic process, for example $f(s) = e^{t-s}$. Is $f$ adapted to $\mathbb F$?
Thank you! :)
A process $f: [0,\infty) \times \Omega \to \mathbb{R}$ is $\mathcal{F}_s$-adapted if, and only if, $$(\Omega,\mathcal{F}_s) \ni \omega \mapsto f(s,\omega)$$ is measurable for any fixed $s \geq 0$, i.e. if
$$\{\omega \in \Omega; f(s,\omega) \in B\} \in \mathcal{F}_s \quad \text{for any Borel set} \, B.$$
For fixed $ \geq 0$, the function
$$\omega \mapsto f(s,\omega) = e^{t-s}, \qquad \omega \in \Omega,$$
is simply a constant (it does not depend on $\omega$). It is widely known that constant functions are measurable with respect to any $\sigma$-algebra; hence in particular with respect to $\mathcal{F}_s$. This already proves that $f$ is $\mathcal{F}_s$-adapted.