# Help Understanding the Lebesgue-Radon-Nikodym Theorem

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!

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.