Why can’t measures be defined on uncountable powersets? Example that actually applies to probability theory What actually breaks in probability when you let $\mathcal F = 2^\Omega$ for uncountable $\Omega$? 
To the best of my knowledge, this is not a duplicate question. While there are similar-sounding questions, answers to them usually cite:


*

*Vitali sets. However, only translation-invariant measures prevent Vitali sets. Probability measures don’t need to be translation-invariant, so Vitali sets are irrelevant. You can totally have a probability measure and $\sigma$-algebra containing Vitali sets.

*Lebesgue measures. Same objection as above. 

*Banach-Tarski. But measures must be at least rotation-invariant to prevent Banach-Tarski (or live in 2-D). Probability measures don’t need to be rotation-invariant, so Banach-Tarski is irrelevant.
AFAICT, there seems to be a logical gap in the usual explanations. 
Could someone provide a concrete example of any paradoxes that emerge from uncountable sets that apply to probability theory? Any issues with letting $\mathcal F = 2^\Omega$? 
Perhaps there is no such paradox. For example, $\delta_a(\cdot)$ seems like a perfectly good probability measure that works with $\mathcal F = 2^\Omega$. The answer might “There’s no paradox, but here is an extra property X we want from our probability measures, here’s why it matters, and for this restricted class of measures, they cannot be defined on $\mathcal F = 2^\Omega$.” If so, I’d love to know what this property $X$ would be.
Apologies if I’ve made an error in this post; I’m new to measure theory!
 A: There's nothing that breaks.  You will certainly almost never see any theorem that requires your $\sigma$-algebra to not have the form $2^\Omega$.  It's just that most naturally occurring measures on uncountable sets are not defined on all subsets (because they are typically ultimately derived from Lebesgue measure, one way or another).  The point of the formalism of $\sigma$-algebras is not to disallow $2^\Omega$, but to allow more general $\sigma$-algebras.
A: You already listed several examples of such properties $X$. They are important because in applications our spaces do have translation and rotation invariance so if the probability measure does not have such properties then it is useless for the application.
For example, intuitively the probability that a uniformly chosen random point in $[0,1]$ lies within the interval $[a,b]$ should be equal to $b-a$. So it depends only on the length and not the position of the interval. To make this precise, we construct a translation-invariant measure on $[0,1]$.
A: I'm two years late, but you might be interested in the following construction. In a discrete, real-world probability space, we can define a notion of translation-invariance, and as a result construct non-measurable sets.
Pick a natural number $n$. Let $\Omega = \{ 1, \dots, n \}^\mathbb{N}$ for some natural number $n$. The one assumption we need for this construction is that each of the outcomes $1, \dots, n$ should be equally likely. This is hopefully fine by you because this model certainly comes up a lot—$n = 2$ is an infinite sequence of coin tosses and $n = 6$ is an infinite sequence of die rolls.
Define an equivalence relation on $\Omega$ as follows. Say $\omega_1 \sim \omega_2$ if they differ in finitely many entries. Then let $S \subseteq \Omega$ consist of one representative per equivalence class.
Note that the set of maps on $\Omega$ which change finitely many entries is countable. So call these maps $\sigma_1, \sigma_2, \dots$. Then, for any $i\neq j$, we have that $\sigma_i(S)$ and $\sigma_j(S)$ are disjoint. Moreover, $ \bigcup_{i} \sigma_i(S) = \Omega$.
By the "outcomes are equally likely" assumption, we know that the $\sigma_i(S)$ all have the same measure. Since the $\sigma_i(S)$ are disjoint, by additivity, we should have $\sum_i \mu(\sigma_i(S)) = \mu(\Omega) $. This implies $\sum_i \mu(S) = 1$, which is a contradiction.
So the physical interpretation is we can't assign probabilities to all possible subsets of sequences of (for example) infinitely many coin tosses.
