Measurability of vector valued functions and the hyper plane separation counterpart Let $([0,1],\Sigma,\mu)$ be a probability space where $P$ is $\sigma$-finite. Define a  product measure $\lambda=\mu\times \mu$ on $([0,1]^2, \Sigma\times \Sigma)$ .
Consider vector valued functions   $ f\in L^2([0,1]^2,\lambda,R^n)$, i.e. $f=(f_1,...,f_n)$.


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*Can we say $f\in L^2([0,1]^2,\lambda,R^n)$ is $\lambda$-measurable, if $|| f,g||:[0,1]\to R$ is $\lambda$-measurable for every  $g\in L^2([0,1],\lambda,R^n)$? Here   $|| .,.||$ is the Euclidean inner product norm in $R^n$.

*Would $f$  be $\lambda$-measurable directly implied by  $f\in L^2([0,1]^2,\lambda,R^n)$?

*Would $f_k(.,y)$   be $\mu$-measurable for every $y\in[0,1]$?

*Would hyperplane separation theorem has the counterpart for the space of vector valued functions?  i.e.
If $f^*\notin S$, where S is a compact and convex subset of  $L^2([0,1]^2,\lambda,R^n)$, then there exists $g\in L^2([0,1]^2,\lambda,R^n)$ such that
$$\langle g,f^*\rangle>\sup_{f\in S} \langle g,f\rangle$$
where $\langle g,f \rangle=\int_{[0,1]^2} ||g(x),f(x)||^2 d\lambda(x)$.
Thanks.

2: Yes, this is part of the definition of the space $L^2$.
3: Not necessarily for every $y$, but at least for $\lambda$-almost every $y$.  This is part of Fubini's theorem.
4: Yes, this is a consequence of the Hahn-Banach theorem and is true in every normed space.
