The multiplier ideals of a sum of ideals

We prove that if $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}\subseteq\mathcal{O}_X$ are nonzero sheaves of ideals on a complex smooth variety $X$, then for every $\gamma\in{\mathbb Q}_+$we have the following relation between the multiplier ideals of $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}$ and $\underline{\mathbf{a}}+\underline{\mathbf{b}}$:

A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals.

We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.