I want to prove following (from Folland, Ex. 3.26): If $\lambda$ and $\mu$ are positive, mutually singular Borel measures on $R^n$ and $\lambda + \mu$ is regular, then so are $\lambda$ and $\mu$.
For definitions, while searching math.SE, I came across to this question which includes the Folland's definition of the regular measure. You can see that question.
What I have tried so far:
Since for a compact $K$, $(\lambda + \mu)(K) < \infty$, and this implies another condition for regularity, I tried to show that if this implies $\lambda(K) < \infty$ and $\mu(K) < \infty$. Since $\lambda$ and $\mu$ is mutually singular, $\exists$ $E,F \in \mathcal{M}$ (suppose they are defined on a measure space $(X,\mathcal{M})$), such that $E \cap F = \emptyset$ and $E \cup F = X$. However, I could not show anything using these properties.
This is a homework question, so reasonable hints are more than welcome. Question seems rather simple, however I am missing something. Thanks!