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

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What is $E$...? – Alon Amit May 21 '11 at 5:42
Sorry, that should be $\mathbb{R}$, the real line. – Hawii May 21 '11 at 5:44
Lovely comments! :) – AD. May 21 '11 at 9:04
This looks like homework. If so, please add the "homework" tag, and read… and edit your question accordingly. If not, please give some motivation for the question, and explain any progress you have made. – Nate Eldredge May 21 '11 at 12:33
This is not a homework problem, and I am not taking any course now. I'm learning some probability theory from GTM 261 "Probability and Stochastics", and this is an exercise on page 46. I'm interested in knowing the answer because this result is used in the chapter on martingales. I think it has to do with either the countability of rational numbers or constructing a sequence of $\mathcal{B}\otimes\mathcal{F}$-measurable functions converging to the function in question. However, I'm stuck in either way. – Hawii May 21 '11 at 21:25

2 Answers 2

up vote 3 down vote accepted

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.

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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!

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