Is it possible to place non-trivial probability measures on sets of cardinality strictly greater than the continuum -- in particular, on sets of cardinality 2^c? (Any references would be appreciated.)



Probability Measures and Cardinality > c
There's even more. If your set can be considered as a product of probability spaces, its measure will have some good properties. This is a theorem due to Alexandra Ionescu Tulcea: given an infinite (with arbitrary cardinality) family of probability spaces
$$\left\{(\Omega\_n, \mathscr{B}\_n, \mu\_n)\right\}\_{n \in I}$$
if we call
$$\pi\_m : \prod\_{n \in I} \Omega\_n \rightarrow \Omega\_m$$
the canonical projection, then it's possible to give $\Omega \mathrel{\mathop{:}=} \prod\_{n \in I} \Omega\_n$ the structure of a probability space with measurable sets
$$\mathscr{B} = \bigotimes\_{n \in I} \mathscr{B}\_n$$
and probability measure $\mu : \mathscr{B} \rightarrow \mathbb{R}$ such that
$$\forall \{n\_1, \ldots, n\_t\} \subset I \ \ \forall (A\_{n\_1}, \ldots, A\_{n\_t}) \in \prod\_{i = 1}^t \mathscr{B}\_{n\_i}$$
$$\mu\left(\pi\_{n\_1}^{-1}(A\_{n\_1}) \cap \ldots \cap \pi\_{n\_t}^{-1}(A\_{n\_t})\right) = \mu\_{i\_1}(A\_{i\_1}) \ldots \mu\_{i\_t}(A\_{i\_t})$$


