I want to compute the POVM $E_{(\theta, \phi)}$ of the measure which gives the probability of a qubit state lying over the surface of Bloch sphere, with angles $\theta, \phi$. How can I handle this?
Dividing the surface area through the volume of Bloch sphere seems not to be a solution because it gives and absurd result of 3.
How should I start?
EDIT_______
For clarification, I will write down the full statement of the problem:
Find the POVM $E_{(\theta, \phi)}$ of the measure which determines the probability that the state of a qubit is on the surface of the Bloch sphere with polar angle $\theta$ and azimuthal $\phi,$ the relation of completeness being fulfilled in this case: $$\int d^{2} \Omega E_{(\theta, \phi)}=\mathbb{I}$$ Is that a projective measure? Prove also that: $$p(\theta, \phi)=\operatorname{Tr}\left[E_{(\theta, \phi)} \rho\right]=\frac{1}{4 \pi}(1+\boldsymbol{r} \cdot \boldsymbol{\Omega})$$ with $r$ being the state's Bloch vector and $\Omega=(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$.
Explain why the only determination of the maximum of $p(\theta, \phi)$ allows tomography the state.
I was trying to use the pure state $\rho = \frac{1}{2}\left( \mathbb{I}+\vec{\Omega } \cdot \vec{\sigma} \right)$ as a operator measure for the mixed state $\rho '$ but can't obtain the factor $\frac{1}{4\pi}$.
I hope this update clears up your doubts
