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

So we have $\mu:{\cal P}(\mathbb{N})\to [0,1]$, and $\mu$ has several nice properties: it is finitely additive and translation invariant.

Can this be extended to $\mathbb{N}^2$? By this I mean the following: Is there a map $\mu_2:{\cal P}(\mathbb{N}^2)\to [0,1]$ that is also finitely additive and translation invariant and so that for all $A,B\subseteq \mathbb{N}$ the following holds? $$\mu_2(A\times B) = \mu(A)\cdot\mu(B)$$