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In my lecture note, a real-valued random variable is defined as follows:

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Then I have an exercise:

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Actually, my lecture notes do not have a definition of a random variable generated by other ones.

We already have funtions $X : \Omega \to \mathbb{R}$ and $Y : \Omega \to \mathbb{R}$.

My question: What is the random variable $Z: \Omega \to \mathbb{R}$ by the symbol $Z= X/Y$? I mean, for each $\omega \in \Omega$, what is $Z(\omega)$?

Thank you so much for your help!

Akira
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    Just prove $X/Y$ is measurable if $X$ and $Y$ are measurable. – Nap D. Lover Sep 27 '19 at 20:48
  • Hi @NapD.Lover, I meant for each $\omega \in \Omega$, how to define $Z(\omega)$? – Akira Sep 27 '19 at 20:53
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    For each $\omega\in \Omega$, define $Z(\omega)=X(\omega)/Y(\omega)$ whenever $Y(\omega) \neq 0$ where $X$ and $Y$ are defined measurable functions on $\Omega$. This defines the function $Z:\Omega\to \mathbb{R}$ provided $Y\neq 0$. For $Z$ to be a RV all thats left is to prove that $Z$ is measurable. Is this clear? – Nap D. Lover Sep 27 '19 at 20:54
  • I'm sorry @NapD.Lover. I'm not clear enough for my question, so I have just edited it. Could you please read it again? – Akira Sep 27 '19 at 20:58
  • What is N* here? –  Sep 27 '19 at 21:03
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    You ask: "what is $Z(\omega)?$". I just told you what it is. For every $\omega$ such that $Y(\omega) \neq 0$, we define $Z(\omega)$ to be the function $Z(\omega)=X(\omega)/Y(\omega)$. What is your confusion? $X$ and $Y$ are already defined functions on $\Omega \to \mathbb{R}$, they are also random variables meaning they are $\mathcal{A}$-measurable i.e. for any Borel set $B$, we have $X^{-1}(B)\in \mathcal{A}$, and $Y^{-1}(B)\in \mathcal{A}$. In order to prove $Z$ is also a random variable you need to prove $Z$ is measurable i.e. for any Borel set $B$, that $Z^{-1}(B)\in \mathcal{A}$. – Nap D. Lover Sep 27 '19 at 21:03
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    @d.k.o. I think it is the set of positive integers. – Akira Sep 27 '19 at 21:04
  • By the way, you might find this post helpful, https://math.stackexchange.com/questions/846011/precise-definition-of-the-support-of-a-random-variable?rq=1 – Nap D. Lover Sep 27 '19 at 21:13
  • @NapD.Lover From your comment, $\mathbb P(Z=z) = \mathbb P({\omega \in \Omega \mid X(\omega)/Y(\omega) = z})$. It seems that I unconsciously assume that $X,Y$ are random variables induced by the same probability space. What if they are induced by different probability spaces? – Akira Sep 27 '19 at 21:24
  • Hi @d.k.o., I just read from here about the sum of r.v and found that it's is different than the way you mentioned about the ratio of two r.v. Could you please have a look at my link? – Akira Sep 27 '19 at 21:49
  • I accidentally deleted my comment. However, consider 2 functions: $f:R\to R$ and $g:R^2\to R$. What is $f+g$ or $f/g$? –  Sep 27 '19 at 21:53
  • @d.k.o., they don't have the same domain, so $f+g$ is undefined. – Akira Sep 27 '19 at 21:54
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    The same happens to $X$ and $Y$ if they have different domains. –  Sep 27 '19 at 21:56
  • @d.k.o. Yeah but can’t we construct $X’, Y’$ on a common probability space given $F_X$ and $F_Y$ such that $X$ and $X’$ are equal in distribution and $Y$ and $Y’$ are equal in distribution? – Nap D. Lover Sep 27 '19 at 22:22
  • We can do that, of course. But why? –  Sep 27 '19 at 22:37
  • @d.k.o. The OP asked how to define the quotient of $X$ and $Y$ if they are given on different sample spaces. Obviously we can’t directly but if we instead study $X’/Y’$, shouldn’t this serve most probabilistic needs? – Nap D. Lover Sep 27 '19 at 22:46

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