Main Question or Discussion Point
Why is it so important that a measure is defined on a sigma-algebra? Which properties of the sigma-algebra are crucial for the properties of a measure? Since a measure is by axiom defined on countable unions of sets, it makes sense that a measure should be defined on a family of sets which preserves this property. But for a sigma-algebra A we also have as axiom that if b is a member of A then bc is also a member of A. Is this significant for some of the properties we want for a measure?