I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent Bernoulli random variables.


If you were teaching such a course and had a list of canonical examples to illustrate definitions and theorems, what would be on the list?



Examples of random variables
That will be not quite an answer for your question. Anyway it may be helpful. 


1. If you have a sequence of independent Bernoulli r.v. $(B\_i)$ then you can define a uniform variable by $U = \sum 2^{-i} B\_i$ and further you can obtain an infinite sequence of independent uniform r.v $U\_i$. (just by splitting $B\_i$ into infinitely many subsequences). Finally from this sequence you may get a sequence of independet variables of any continuous distribution $F^{-1}(U\_i)$ where $F^{-1}$ is the generalised inverse of the cdf of the distribution we want to have.
2. There is a construction of the Wiener process using Haar functions. I guess it is may be to difficult for your students. But I can look for references in English (in a moment I have only a Polish book).


