Consider the measurable space $2^{\mathbb R}$, equipped with the tensor-product $\sigma$-algebra. Famously, this space has a measurable structure which is not generated by a topology (see [this answer](https://mathoverflow.net/a/87888/238)).


Can you provide an example of a non-trivial probability measure on $2^{\mathbb R}$?



A non-trivial probability measure on $2^{\mathbb R}$
$2^{\mathbb{R}}$, being a product of compact Hausdorff groups, is a compact Hausdorff group, so it has a normalized Haar measure ("flipping uncountably many coins"). 


