for any two positive integers consider the set . Such sets are measurable and is a union of such sets
Let be a sequence of measurable sets in . Let be a subset of so that if and only if belongs to exactly 2 of the sets . Determine whether E is measurable. If it is, show it. If not, give a counterexample.
I suppose it is measurable? No? Can't think of any counterexample.
Thanks in advance.
In my opinion any question about "something" should start off with a definition of "something" and a demonstration that the definition is satis fied. So here is my little contribution as a matter of principle from http://en.wikipedia.org/wiki/Measure_(mathematics)
σ-algebra, meaning that unions, intersections and complements of sequences of measurable subsets are measurable.
Let Σ be a σ-algebra over a set X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:
for all
- Non-negativity:
- Countable additivity (or σ-additivity): For all countable collections of pairwise disjoint sets in Σ:
- Null empty set:
I would find it interesting to see how this would be done in the case of the previous post.