I am reading signed measures and related topics for the first time and am naturally a little confused. Suppose if I have a measurable space $(X,\mathcal{M})$ and two finite positive measures on it, namely $\mu_1$ and $\mu_2$. Then $\nu=\mu_1-\mu_2$ is a finite signed measure on $(X,\mathcal{M})$. Now if I take my reference positive measure as $\mu_1$ and apply the LRN theorem, it states that there exist unique signed measures $\lambda$ and $\rho$ such that $\nu=\lambda+\rho$, $\lambda \perp \mu_1$ and $\rho \ll \mu_1$. My confusion is that since $\nu=\mu_1-\mu_2$ and $\mu_1 \ll\mu_1$ trivially, this would mean that $\lambda=-\mu_2$ and that $-\mu_2\perp\mu_1 \iff \mu_2\perp\mu_1$. But this is not necessarily true for example if I take $X=[0,1],\mathcal{M}=\mathcal{B}_{[0,1]}$ and $\mu_1=\mu_2=m$ (the Lebesgue measure).
Clearly I am wrong and certainly missing a basic understanding. Any help therefore will be greatly appreciated. Thank you!
Best Answer
Since $\mu_2\perp\mu_1$ is not true, $-\mu_2$ is not the $\lambda$ given by the LRN theorem. You must satisfy all the conditions of the theorem in order to be unique. Satisfying 2 of 3 conditions does not guarantee uniqueness.