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 $(\prod\_{n \in \mathbb{N}} \mathbb{R},B)$,
* Locally-positive $\sigma$-finite (but not finite) Borel measures on$(\prod\_{n \in \mathbb{N}} \mathbb{R},B)$?


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



Explicit examples of (probability) measures on $\prod \mathbb{R}$
$\newcommand\R{\mathbb R}$ Let $T$ be any countable nonempty set. Let $B$ be the Borel $\sigma$-algebra over $\R^T$ generated by the Tikhonov product topology on $\R^T$. Take any $t\_0\in T$. For each natural $k$ and each $t\in T$, let 
$$\nu\_{k,t}:=
\begin{cases}
N(0,1)&\text{ if } t\ne t\_0,\\
N(k,1)&\text{ if } t=t\_0. 
\end{cases}
$$


By [Kolmogorov's measure extension theorem](https://en.wikipedia.org/wiki/Kolmogorov_extension_theorem#General_form_of_the_theorem) , for each natural $k$ there is a product probability measure 
$$\mu\_k:=\bigotimes\_{t\in T}\nu\_{k,t} $$
on $B$, which will obviously be locally positive. 


Moreover,  

$$\mu:=\sum\_{k=1}^\infty\mu\_k$$
will be a locally-positive $\sigma$-finite (but not finite) measure on $B$. 


The same constructions will work for uncountable sets $T$ if, instead of the Borel $\sigma$-algebra $B$, we will take the $\sigma$-algebra generated by the standard base of the Tikhonov product topology. 


