reference request – Dimension of cartesian products

Is there a notion of dimension such that for all Borel sets $A,Bsubseteqmathbb{R}^{n}$ we have
$$ dim(Atimes B)=dim(A)+dim(B)?$$ For topological, Minkowsky, packing and Hausdorff dimension this is not true. If the answer is no (which I suppose) I like to understand which problems appear, if we try to give such a defintion of dimension.