Measurable Spaces and Functions
Theorem:
nonexistence of extension of length to all subsets of
\@{nonexistence-of-extension-of-length-to-all-subsets-of-reals}
There does not exist a function with all of the following properties:
- (a) is a function from the set of subsets of to
- (b) for every @open interval of
- (c) for every @disjoint sequence of subsets of
- (d) for every and every
Proof
\@{proof-of-nonexistence-of-extension-of-length-to-all-subsets-of-reals}
Outline: assume such a exists. Then we can show that it has all of the properties of outer measure that were used to prove Nonadditivity of outer measure.