Let ($\Omega, \mathcal A, P)$ be a measure space, and $]0,1[$ equipped with the corresponding Lebesgue $\sigma$-algebra and Lebesgue measure.
We have a function $f:]0,1[ \times \Omega \to \mathbb R$
What does mean $$\Vert f \Vert_{L^2(]0,1[\times \Omega)}$$
And is it the same as (or does this expression make sense) $$\Vert f \Vert_{L^2(\Omega\times ]0,1[)}$$
$$\lVert f\rVert_{L^2(]0,1[\times\Omega)}=\sqrt{\int_{]0,1[\times\Omega} \lvert f\rvert^2\,dx\otimes dP}$$ where $dx\otimes dP$ indicates the product measure on the product $\sigma$-algebra of $]0,1[\times\Omega$. Namely, the product $\sigma$-algebra is the one generated by products of measurable sets. The product measure is the only measure on that $\sigma$-agebra such that $\mu(A\times B)=\mathcal L(A)\times P(B)$ for all $A,B$ measurable in the original spaces (with the notational assumption that $0\cdot M=0$ for all $M$ including $\infty$).
The other thing does not make sense.