I am currently preparing for a measure theory exam and struggling with the following problem:
For which $p \in \mathbb R$ does a signed measure on $([0, 1], \mathcal B([0, 1]))$ exist such that $\mu([0, x]) = x^p \sin(1/x)$?
Furthermore, it is required to determine a Hahn-decomposition of $\mu$.
So far I have tried differentiating $x^p \sin(1/x)$ and then analyzing where the derivative, call it $f$, is positive and negative (for the Hahn-decomposition). Then one might set $\mu(A) = \int_A f \mathrm d \lambda$. I have done this in similar exercises. However, what bothers me is the point $0$. But maybe the worry is not necessary, since ignoring single points does not hurt when finding densities with respect to the Lebesgue measure $\lambda$.
Regarding which $p \in \mathbb R$ is permissible, I only know that from $\mu([0, 0]) = 0$ it follows that $p > 0$.
I'd appreciate if somebody could back up my thoughts or provide an alternative strategy.