Due May 11
Let be a rotation. Then it induces a measure-preserving map of the sphere with its measure .
Use the polar coordinate formula to prove the following statements.
- for any d.
If is a finite Borel measure on the interval , then
is a linear functional on , positive in the sense that if .
Conversely, if l is a positive linear functional on , there exists a unique finite Borel measure on such that for every .