# Thread: probability space and non-random variables

1. ## probability space and non-random variables

Hello! on my book there is this exercise, but I don't understand what it is asking and clearly I don't know how to solve it.

It says: Give an example of a probability space and a real-valued function on Ω= {0,1,2} that is not a random variable.

2. Just to get a feel on what level this is, what is your definition of a random variable?

3. well I'm just trying to come up with an algebra A and a function X such that there exists a B Borel for which X^-1 does not belong to A.
But I can't make up one
Any ides?

4. How about equipping your set $\displaystyle \Omega=\{0,1,2\}$ with the trivial sigma-algebra, containing only all of $\displaystyle \Omega$ and the empty set.

Then whenever you have a map from $\displaystyle \Omega$, it won't be a random variable if the pre-image of a measurable set is not the entire $\displaystyle \Omega$ or the empty set.

5. you are basically saying that I should be taking the case where A is made of omega and the empty set; so that basically A is not a measurable sigma-algebra and so I cannot get X r.v.
did I get it correctly?

6. I am not sure what you mean by a "measurable sigma-algebra". But you are right in that I meant A to consist of both Omega and the empty set. Then most maps X into the real numbers (or other measure spaces) will not be measurable.

7. Well what about if I build the A={emptyset, 1, 2, {1U2}, Omega}
what would be X such that the counter image is not mapping into A ?

8. You could define X : Omega -> R by X(1) = 1, X(2) = 2, X(3) = 3.

Then take an open interval I in R that contains 3, but not 1 or 2. Then the preimage of I under X is {3}, which is not in the sigma-algebra.