Currently, I am reading "Interpolation Theory" (3rd edition) by Alessandra Lunardi. In Example 1.15, the following is claimed: A function $$ w \colon (0,\infty) \to L^p(\mathbb R^n) $$ is Bochner measurable if and only if the function $$ w_{t,x} \colon (0, \infty) \times \mathbb R^n \to \mathbb R, (t,x) \mapsto w(t)(x) $$ is measurable.
So far, I have achieved to prove the direction "$\Leftarrow$". However, I don't get the other direction working. Here is the best I have done so far: Since $w$ is Bochner measurable, there is a sequence of simple functions converging point-wisely to $w$, i.e. there are disjoint Borel measurable sets $A_{i,n}$ and $a_{i,n} \in L^p(\mathbb R^n)$ such that for $$ f_n = \sum_{i=1}^{m_n} a_{i,n} \chi_{A_{i,n}} $$ we have that $f_n(t) \to w(t)$ in $L^p(\mathbb R ^n)$ for every $t > 0$. In order to get measurability of $w_{t,x}$ I would hope for the point-wise convergence of $f_n(t)(\cdot) \to w(t)(\cdot)$, however I have no idea how to achieve this for all $t$ simultaneously (since there are uncountable many $t$ a diagonal sequence argument does not seem to work).
Any kind of help or hint is appreciated.