Hausdorff decomposition of a measure

Question

Let $\mu$ be a probability measure on $(\mathbb R^n,\mathcal B^n)$. Is it true that $\mu$ can be decomposed as a countable sum of measures which are only distributed on sets of a fixed Hausdorff dimension? $$$$ In formal terms that $$\forall A\in \mathcal B^n,\ \ \ \mu(A)=\sum_{k=1}^\infty \mu_{a_k}(A)$$ s.t. for all k $$\exists E,\ dim_H(E)=a_k:\ \ \mu_{a_k}(\mathbb R^n-E)=0$$ $$\forall F,\ dim_H(F)<a_k:\ \ \mu_{a_k}(F)=0$$ Note that the result trivially holds for all continue probability distributions and for all discrete ones. I cannot imagine a probability measure $\mu$ such that it does not hold and since it is rather interesting and useful I wonder if it plays a role in some branch of measure theory.