Lecture
This lecture covers the construction of measures in the context of Hilbert spaces, focusing on positive functionals. It explores extreme cases of o-algebras with measures, such as the Dirac measure. Positive functionals on continuous functions with compact support lead to Borel o-algebras and regular, complete, and locally finite measures. The lecture defines interior and exterior measures, highlighting their properties and relationships. It discusses the concepts of sur-additivity and sub-additivity for these measures, providing proofs and examples. The importance of o-additivity for measures is emphasized, aiming for equality rather than inequality in the context of measure theory.