Does anybody know a specific function $\mu$ going from the power set of the integers to $[0,1]$ so that
1). $\mu(\mathbb{Z}) = 1$
2). $\mu(A\cup B) = \mu(A)+\mu(B)$ if $A$ and $B$ are disjoint.
3). $\mu(A+1) = \mu(A)$ where $A+1 := \{a+1 | a \in A\}$?
I've seen how to show one exists but I was wondering if someone could tell me a specific $\mu$ that works, or why it is impossible to construct explicitly such a $\mu$.
Despite the fact the you mention that you know about how to prove that such a measure exists, I will give some links, where measures of the following form (or very similar) are mentioned: $$\mu(A)=\operatorname{\mathcal U-lim} \frac{A\cap[-n,n]}{2n+1}.$$ (Here $\mathcal U$ is any free ultrafilter. Some of these posts deal with measures on $\mathbb N$ rather than $\mathbb Z$, but it is not difficult to modify them.)
It might be also worth mentioning that existence of a shift-invariant finitely additive probability measure is equivalent to existence of shift-invariant mean.
In some the linked posts you can find some further references. In particular, van Douwen's paper and the book by Howard and Rubin are certainly worth having a look in connection with this.
You cannot expect a very explicit example - since this is not provable in ZF. This is true already if you require the first two conditions. You can find some references here:
