Measure Theory 12 | Carathéodory's Extension Theorem [dark version]

[@benjaminlloyd-hughes5621]

Julian, you're amazing! Thanks to you I think I can answer my own question, but I would love to hear your words on "...and maybe we will talk about the proof later."

[@tomkerruish2982]

Property (2) of a semiring is redundant, as A\(A\B) equals the intersection of A and B. Edit: Come to think of it, so is property (1), as A\A = Ø, although we would still need to specify that a semiring must be nonempty.

[@brightsideofmaths]

To be honest, I don't understand your comment. Property (1) and (2) tell which sets are elements of the semi-ring and property (3) says something about how differences look.

[@tomkerruish2982]

@brightsideofmaths What I mean is that, if you have a family of sets which satisfies property (3) (closure under set difference), then it necessarily satisfies property (2) (closure under set intersection). Additionally, any nonempty family of sets closed under set difference must contain the empty set, as A\A = Ø for any set A in the family (we must assume the family to be nonempty in order to justify selecting a member set A).

[@brightsideofmaths]

@@tomkerruish2982 "Closure under set differences" is not what property (3) says.

[@isaacagyei3952]

Thank you so much for such incredible material. Please, could you give me a lecture on "Absolute continuity of measures, that's Signed measures and Absolute continuity"?. If you already have them, kindly let me know. Thank you