Suppose I have $3$ sets $A,B,C$ all contained in let's say $\mathbb{R}^k$ and I have the relation that with respect to some dimension function $f$
$H^f(A \cap B)= H^f(A \cap C)$
where $H^f(A)$ denotes the usual Hausdorff $f$-measure of the set $A$.
Is it true that
$H^f(B)= H^f(C)$.
If not what extra conditions are required for this conclusion?