Let $C\left[0,T\right]$ be the space of continuous functions from $\left[0,T\right]$ to $\mathbb{R}^{K}$, endowed with the topology of uniform convergence. When we say that $f:C\left[0,T\right]\rightarrow\mathbb{R}$ is a measurable functional, we mean measurable with respect to the Borel $\sigma$-field of $C\left[0,T\right]$.
Assume $\left\{Z_{t} : 0 \le t \le T \right\} $ is a $\mathbb{R}^{K}$-valued continuous process and the $\sigma$-field $\mathscr{F}_{t}$ is generated by $\left\{ Z\left(s\right):0\le s\le t\right\}$ and assume $\mathscr{F}=\mathscr{F}_{T}$. If $X$ is $\mathscr{F}_{T}$-measurable random variable, I need to show that $X$ has the form $X=f\left(Z\right)$ for some measurable functional $f:C\left[0,T\right]\rightarrow\mathbb{R}$. But I do not know what to do. Can anyone help me?
Hints: for any $k \geq 1$ and points $t_1,t_2,...,t_k$ and and any Borel set $E$ in $\mathbb R^{k}$ the function $g\in C[0,T] \to I_E(g(t_1),g(t_2))..,g(t_k))$ is measurable. If you call this $f$ then $f(Z)=I_{(Z_{t_1},...,z_{t_k})^{-1} (E)}$. This proves the result when $X$ is of the type $I_{(Z_{t_1},...,z_{t_k})^{-1} (E)}$. Now take linear combinations to prove this for simple functions $X$ and the take limits, etc.