Consider a metric space $A$ with a metric $d$, and consider the measurable space $(A,\mathcal{B}(A))$ with the Borel $\sigma$-algebra generated by $d$-open sets. Let $(\Omega,\mathcal{F})$ be a measurable space. Consider the product $\sigma$-algebra $\mathcal{B}(A)\otimes\mathcal{F}:=\sigma(\{B\times F\mid B\in\mathcal{B}(A),\,F\in\mathcal{F}\})$
Suppose a function $f:A\times\Omega\to \mathbb{R}$ satisfies
- $f(\cdot,\omega):A\ni a\mapsto f(a,\omega)$ is a continuous function for each $\omega\in\Omega$.
- $f(a,\cdot):\Omega\ni \omega\mapsto f(a,\omega)$ is a $\mathcal{B}({A})$ measurable.
Question: Can we say that $f$ is $\mathcal{B}(A)\otimes\mathcal{F}$-measurable?
My guess is we can. I am aware that there are similar questions where $A:=\mathbb{R}$, e.g.,
A function which is continuous in one variable and measurable in other is jointly measurable
A question on measurability in product spaces
A question regarding separable continuity and measurability
and I think what I should do is to something in line with constructing a sequence $f_n$, step function in $a\in A$, and whose pre-image is a rectangle, then use the continuity in $a$. But I am not really sure how to write down explicitly.
Best Answer
No. The following example is due to R. O. Davies, Separate approximate continuity implies measurability, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 73, Issue 03, May 1973, pp 461.
Let $\kappa$ be a real valued measurable cardinal and $m: \mathcal{P}(\kappa) \to [0, 1]$ be a $\kappa$-additive diffused probability measure. Put discrete metric on $\kappa$. Then the characteristic function of $\{(\alpha, \beta) : \alpha < \beta < \kappa\}$ is continuous in each coordinate but not $m \otimes m$-measurable. This easily follows from Fubini's theorem.