In this paper, Robert Aumann claim that(page 508):
But as shown at the bottom of page 520, all these sample spaces don't admit uncountable independent random variables.
What's the implication of this? In "real life" life, we don't need uncountable independent random variables, or Aumann's list of sample spaces is incomplete?
Added: Thank the user called Mars for pointing out that I should provide enough context. Indeed, $I$ is the unit interval, $[0,1]$. It seems "real life" is referring to two-person zero-sum sequential games with denumerable stages.
Added${^2}$: At section 9, Aumann explains that why he don't let $b_i( \cdot, x)$ and $b_i( \cdot, x')$ be independent, that is, given two different realizations, $x$ and $x'$ of information set at the stage stage $i$, the random variables whose values is in the action space of stage $i$ needn't be independent. It seems that his reasoning goes like the above mentioned three sample spaces don't admit uncountable independent random variables. To fit his definition of behaviour strategy into this framework, we have to forsake the independence of $b_i( \cdot, x)$ and $b_i( \cdot, x')$. This is weird to me. Why not find a larger sample space that admits uncountable independent random variables? Is there any other reason?
First, your question didn't provide enough context. I glanced at the paper in order to try to figure out what's being asked, but that shouldn't be necessary, in general. The question should provide sufficient context on its own. e.g. What is I? It turns out it's the unit interval (p. 507); that's important information here. For the question of what Aumann means by "real life", it matters that he's talking about two-person games with mixed strategies.
Here's a guess at an answer, based on quickly skimming a few paragraphs: On page 508, Aumann's saying there that there are strategies each of which involves continuous variation in one dimension. Each such strategy must be represented by a continuous random variable, i.e. one with an uncountable number of possible states (but for his purposes we might as well represent that using the unit interval--nothing will be lost if we do). On page 520 he's talking about the number of such strategies---the number of such random variables. He's arguing that only a countable number of such r.v.'s are needed.