Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some concrete examples of:

• Locally-positive Borel probability measures on $(\prod_{n \in \mathbb{R}} \mathbb{R},B)$,
• Locally-positive $\sigma$-finite (but not finite) Borel measures on$(\prod_{n \in \mathbb{R}} \mathbb{R},B)$?

How does the situation change when the product is indexed over $\mathbb{R}$?