Can you give an example of some Boolean algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive extension to the $\sigma$-algebra generated by this algebra, and moreover, such that when shifting any set $A ∈ \mathcal A$ by an integer $n$ (calling the resulting set $A + n$), the following equation is fulfilled: $A + n ∈ \mathcal{A}$, $\mu (A + n) = \mu (A)$?

$\mathcal F$ is set of events. Can you give an example of some algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive extension to the $\sigma$-algebra generated by this algebra, and moreover, such that when shifting any set $A ∈ \mathcal F$ by an integer $n$ (calling the resulting set $A + n$), the following equation is fulfilled: $A + n ∈ \mathcal{A}$, $\mu (A + n) = \mu (A)$?

Can you give an example for some algebra $\mathcal A$ over $\mathbb N$ a non-zero finite additive measure $\mu $ on this algebra, which has a countably additive extension to the $\sigma$-algebra generated by this algebra, moreover, when shifting any set $A ∈ \mathcal F$ by an integer $n$, for the so obtained set $A + n$ was fulfilled: $A + n ∈ A$, $\mu (A + n) = \mu $(A)?

Can you give an example of some Boolean algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive extension to the $\sigma$-algebra generated by this algebra, and moreover, such that when shifting any set $A ∈ \mathcal A$ by an integer $n$ (calling the resulting set $A + n$), the following equation is fulfilled: $A + n ∈ \mathcal{A}$, $\mu (A + n) = \mu (A)$?