Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space, and define $(L_2[0,1],\mathcal{B}(L_2[0,1])$ be the Borel measure space of $L_2([0,1])$ with respect to the standard $L_2$ norm. Define a sequence of random variables $c_k$ for $1\leq k \leq M$ for some fixed $M>0$. Then if $\{x_1,...,x_m\}$ are equally spaced points on $[0,1]$, then the random variables $(c_k)$ define a random step function $g$, such that $g(\omega,x) = c_k(\omega)$ for $x\in[x_{k},x_{k+1})$, $\omega\in\Omega$ (basically, for every $\omega \in \Omega$, we define a sample path in $L_2[0,1]$).
We now define a random element: $X: \Omega \mapsto L_2([0,1])$ is a random element if for all $B\in \mathcal{B}(L_2([0,1]))$, then $X^{-1}(B) \in \Sigma$.
Question: Is my random step function $g$ a random element from $(\Omega,\Sigma,\mathbb{P})$ to $(L_2[0,1], \mathcal{B}(L_2[0,1]))$?
My intuition believes that I am correct, since $L_2[0,1]$ is a Banach space with respect to the $L_2$ norm, and so Wikipedia (https://en.wikipedia.org/wiki/Random_element) says that so long as every linear bounded functional applied to $X$ is a random variable (which it is for our function g) then $X$ is a random element, although I am not convinced by this statement. Would anyone happen to know how to show that for every closed ball $B$ in $L_2[0,1]$, then $\{\omega\in\Omega | g(\omega)\in L_2[0,1]\} \in \Sigma$?
EDIT 1: I forgot to mention that here, we assume that $\sup_{1\leq k\leq m}|c_k| \leq C_0$ almost surely, for some constant $0<C_0<\infty$, and so $g$ is square integrable almost surely.
EDIT 2: I think the proof would have to do with the fact that each $c_k$ is a Random Variable with respect to $(\Omega,\Sigma,\mathbb{P})$, and there are only finitely many $c_k$, but that is as far as my intuition extends.
There's quite a clean answer to this problem indeed. Since $L_2[0,1]$ is a Hilbert space with respect to the inner product $\langle f,g\rangle = \int_{[0,1]} fg$, Theorem 7.1.2 of Hsing & Eubank (2015, http://onlinelibrary.wiley.com/book/10.1002/9781118762547) states that any random element $\chi$ from $(\Omega,\Sigma,\mathbb{P})$ to $(L_2[0,1],\mathcal{B}(L_2[0,1]))$ is measurable if for all $h\in L_2[0,1]$, $\langle \chi, h\rangle$ is measureable with respect to $(\Omega,\Sigma,\mathbb{P})$ mapping into $(\mathbb{R},\mathcal{B}(\mathbb{R}))$.
Since our function $g$ is a step function, we have that for any $h\in L_2[0,1]$,
$$ \langle g,h\rangle = \sum_{k=1}^{M}c_k\langle 1_{[x_{k},x_{k+1})},h\rangle, $$
which is clearly measurable since $c_k$ is measurable. Hence $g$ is a random element.