A question on measurability in product spaces

Question

Let $(\mathbb{R},\mathcal{B})$ be the real line with its Borel $\sigma$-algebra and $(F,\mathcal{F})$ be an arbitrary measurable space. Let $f:\mathbb{R}\times F\rightarrow \mathbb{R}$ be such that $y\mapsto f(x,y)$ is $\mathcal{F}$-measurable for each $x\in \mathbb{R}$ and that $x\mapsto f(x,y)$ is right-continuous for each $y\in F$. Show that $f$ is measurable with respect to the product $\sigma$-algebra $\mathcal{B}\bigotimes\mathcal{F}$.

Answer 1

Hint: The right continuity in the $x$ variable gives $f(x,y)=\lim_n\ f(\lceil n x\rceil/n,y).$ This expresses $f$ as the pointwise limit of ${\cal B}\otimes {\cal F}$ measurable functions.

Answer 2

I am working through Cinlar's Probability and Stochastics too, this was my attempted solution, in the spirit of Byron's:

$x\mapsto f(x,y)$ right-continuous for each $y\in F$ implies that we can represent $f$ as the limit of a sequence of functions $f_n$ given by:

$$f_n(x,y) = \sum_{j=1}^\infty 1_{\frac{j-1}{n} \le x < \frac{j}{n}} f(\frac{j}{n}, y)$$

where $1$ denotes the indicator function.

Then $y\mapsto f(x,y)$ $\mathcal{F}$-measurable for each $x\in \mathbb{R}$ implies that $(x,y) \mapsto 1_{\frac{j-1}{n} \le x < \frac{j}{n}} f(\frac{j}{n}, y)$ is ${\cal B}\otimes {\cal F}$ measurable.

Then by Theorem 2.15 in the book, (limit of sequence of measurable functions measurable), $f(x,y)$ is jointly measurable.

Hope this is helpful!