5
$\begingroup$

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 $\mathcal{B}\bigotimes\mathcal{F}$.

$\endgroup$
5
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
1
$\begingroup$

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!

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.