For $A\subseteq \mathbb{N}$ we define the upper density by $$\mu(A)=\limsup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$


A nice property of this map $\mu:{\cal P}(\mathbb{N})\to [0,1]$ is that it is translation-invariant (that is for $A\subseteq \mathbb{N}$ we have $\mu(A) = \mu(n+A)$ for all $n\in \mathbb{N}$, where $n+A = \{n+a:a\in A\}$.


Can this be extended to $\mathbb{N}^2$? By this I mean the following: Is there a map $\mu\_2:{\cal P}(\mathbb{N}^2)\to [0,1]$ with the following properties?


1. $\mu\_2$ is translation-invariant in $\mathbb{N}^2$,
2. for all $A,B\subseteq \mathbb{N}$ we have $\mu\_2(A\times B) = \mu(A)\cdot\mu(B)$, and
3. for all $U\subseteq \mathbb{N}^2$ we have $\mu\_2(U) = \mu\_2(\text{tr}(U))$ where $\text{tr}(U) = \{(y,x)\in\mathbb{N}^2: (x,y)\in U\}$.


Is there a unique choice for $\mu\_2$?



Density measure on $\mathbb{N}^2$
Write $p\_1$ and $p\_2$ for the two projections from $\mathbb N\times \mathbb N\to \mathbb N$. 


Define $\mu\_2(U) = \mu(p\_1[U]) \cdot \mu(p\_2[U])$. This satisfies your demands. 


But this is far from unique. You do not demand that $\mu\_2$ is monotone, so you might also define $\mu\_2(U)=0$ whenever $U$ is not of the form $A\times B$. 


