Is it possible to assign probability to a set $X$ with $|X|>2^{\aleph_0}$? Is it possible to assign probability to a set $X$ with cardinality $|X| > 2^{\aleph_0}$? Example would be a set $|X| = 2^{2^{\aleph_0}}$.
 A: If you wonder why so far only the "trivial" example of a Dirac measure has been mentioned, there is actually a good reason for this. The young Ulam wondered whether there is an uncountable set $X$ for which there is a probability measure
$$
\mu \colon \mathcal P(X) \rightarrow [0,1]
$$
that vanishes on singletons (i.e. $\mu(\{x\})= 0$ for all $x \in X$).
He realized that even in the simpler case that such a $\mu$ is only 2-valued (so for all $Y \subseteq X$ either $\mu(Y) = 0$ or $\mu(Y) = 1$), we cannot prove in ZFC that such an $(X,\mu)$ exists. More precisely: Let $\kappa$ be the smallest cardinal of such an uncountable $X$. Then $\kappa$ is what we nowadays call a measurable cardinal. As such it is inaccessible and thus $V_\kappa \models \text{ZFC}$. By Gödel's incompleteness theorem, we therefore cannot prove that such a $\kappa$ exists (if ZFC is consistent).
Assuming the existence of a measurable cardinal allows us to prove results, that cannot be settled in ZFC alone (e.g. analytic determinacy) and in the last (5?,6?, even more?) decades much work has been done under the assumption of measurable (or other large) cardinals. 
A: Probability is not just defined on a set. It's also defined on a $\sigma$-algebra of subsets. Namely, a probability structure is made of three parts:


*

*The underlying set, say $X$.

*The sets to which you can assign probability.

*The function which assigns probability.


There are several axioms which we require of these structures, such as the sets to which you assign the probability will form a $\sigma$-algebra, and that $\mu(X-A)=1-\mu(A)$ whenever $A$ is a set that can be assigned probability.
Naively we might want to be able to assign probability to every subset of $X$, but even in the case of $[0,1]$ with the usual probability, this is not always possible in the presence of the axiom of choice. More specifically, in the "classical" settings of probability, you can show that there are subsets of $[0,1]$ that you cannot assign a probability, if you want the probability of an interval to be its length.
Which is why we allow some sets not to have a probability. Some trivial possible ways to assign probability is to assign $0$ to any countable subset of $X$, and $1$ to any set $A$ which is co-countable, namely $X-A$ is countable. Others, as suggested are the Dirac measures which concentrate on a singleton, or some particular subset.
A: For any cardinal $\kappa$, you can define the product measure on $2^{\kappa}$ where $0, 1$ are assigned one half measure each. This measure is defined on the sigma algebra generated by clopen subsets of $2^{\kappa}$. Let us call the corresponding measure algebra $\text{Random}(\kappa)$. The class of measure algebras thus obtained forms the building block for every atomless (probability) measure algebra in the following sense:
Theorem (Maharam): Every atomless (probability) measure algebra can be decomposed into countably many pieces each of which is isomorphic (modulo rescaling) to $\text{Random}(\kappa)$ for some $\kappa$.
A: In general for any set $A$ is possible give the probability of dirac, $i.e$ let $x\in X$ and $U\subset A$ we define $\delta_x(U)=1$ if $x\in U$ and $\delta_x(U)=0, \, else.$  so Always is possible give a measure of probability of any set.
