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).
Can you provide an example of 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").
Why can't you just take the product measure induced by the uniform measure on $\{0,1\}$ (or indeed by any other nontrivial measure on this two-element set)? I suppose my question really is: What exactly do you mean by non-trivial?
Instead of giving specific examples of probability measures on $2^{\mathbb{R}}$, I am going to give a couple characterizations of the tensor product $\sigma$-algebra on $2^{\mathbb{R}}$.
If $X$ is a topological space, then recall that set of the form $f^{-1}[\{0\}]$ for some continuous $f:X\rightarrow\mathbb{R}$ is called a zero set. The Baire $\sigma$-algebra on a topological space $X$ is the $\sigma$-algebra generated by the zero sets.
It turns out that if we give $2^{I}$ the product topology, then $2^{I}$ is a compact space and the Baire $\sigma$-algebra on $2^{I}$ is precisely the tensor product $\sigma$-algebra. In fact, if $X$ is a compact totally disconnected space, then the Baire $\sigma$-algebra on $X$ is generated by the clopen subsets of $X$. Therefore, by the Riesz representation theorem, the real-valued measures on the tensor product $\sigma$-algebra on $2^{I}$ are precisely the linear functionals on $C(2^{I})$. Of course, $C(2^{I})$ is the Banach space of continuous functions $2^{I}\rightarrow\mathbb{R}$. As a special case of this fact, the probability measures on the tensor product $\sigma$-algebra on $2^{I}$ are in a one-to-one correspondence with the positive linear functionals on $C(2^{I})$ (i.e. the functions $L:C(2^{I})\rightarrow\mathbb{R}$ with $L(f)\geq 0$ whenever $f\geq 0$).
Suppose that $X$ is a compact totally disconnected space. Let $B(X)$ denote the Boolean algebra of clopen subsets of $X$. Then the Baire probability measures on $X$ can be put into a one-to-one correspondence with finitiely additive probability measures on $B(X)$ as follows: If $\mu$ is a Baire probability measure on $X$, then $\mu|_{B(X)}$ is a finitely additive measure on $B(X)$. If $\nu$ is a finitely additive measure on $B(X)$, then we may extend $\nu$ to a Baire measure on $X$ by Caratheodory's Extension Theorem.
In particular, the probability measures on the tensor product $\sigma$-algebra on $2^{I}$ are in a one-to-one correspondence with the finitely additive probability measures on $B(2^{I})$. It should be noted that the Boolean algebras of the form $B(2^{I})$ are precisely the free Boolean algebras. Therefore, finding probability measures on the tensor product $\sigma$-algebra on $2^{I}$ simply amounts to finding finitely additive probability measures on free Boolean algebras.
