Let $\mathcal{F}$ be a set algebra (or a Boolean algebra). Following Kalton, let me call a function $f\colon \mathcal{F}\to \mathbb R$ $\delta$-additive ($\delta \geqslant 0$), whenever $f(\varnothing) = 0$ and

$$| f(A) + f(B) - f(A\cup B) | \leqslant \delta$$

as long as $A\cap B=\varnothing$ for $A,B\in \mathcal{F}$. Surely 0-additive functions are nothing but finitely additive signed measures.

I am interested in the notion of a tensor product that would be analogous to a product measure but actually only in a very simplistic setting.

Let $X$ and $Y$ be finite sets and suppose that $f\colon \wp(X)\to \mathbb R, g\colon \wp(Y)\to\mathbb R$ are 1-additive functions. Is there a function $h\colon \wp(X\times Y)\to \mathbb{R}$ such that

• $h$ is 1-additive,

• $h(A\times B) = f(A)\cdot g(B)\quad (A\subset X, B\subset Y)$.

The problem is, I think, non-trivial as we need some sort of a canonical decomposition of any given set into a union of rectangles, which is highly non-unique. On the other hand, working only with singletons (trivial rectangles) is not good enough to retrieve the tensorial property of $h$.