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Results tagged with measure-theory
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user 6514
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2
votes
1
answer
207
views
Expanding Measurable Sets
Let $S,T \subset \mathbb{R}^n$ be measurable sets, and suppose that there exists a measurable bijection $f\colon S\to T$ so that
$$
\|f(x)-f(y)\| \;\geq\; \|x-y\|
$$
for all $x,y \in S$. Does it foll …
- 8,483
25
votes
Existence of a strange measure
Just about any form of the axiom of choice can be used to prove this. I like topology, so here's a proof using Tychonoff's theorem.
Consider the space $2^{2^\mathbb{N}}$ of all functions $2^{\mathbb …
- 8,483
21
votes
2
answers
916
views
Codimension of Measurable Sets
I am currently teaching an advanced undergraduate analysis class, and the following question came up.
Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the powe …
- 8,483
10
votes
Accepted
Are finitely generated amenable groups positively finitely generated?
The answer is already no for $\mathbb{Z}$, assuming the question is whether this holds for every meaure. Let $n\in\mathbb{N}$, and let
$$
S \,=\, \{(a_1,\ldots,a_n)\in\mathbb{Z}^n \mid \gcd(a_1,\ldot …
- 8,483