Intereting Posts

Find the center of the group GL(n,R) of invertible nxn matrices.
Why is it that I cannot imagine a tesseract?
Expressing the solutions of the equation $ \tan(x) = x $ in closed form.
Find the maximum and minimum radii vectors of section of the surface $(x^2+y^2+z^2)^2=a^2x^2+b^2y^2+c^2z^2$ by the plane $lx+my+nz=0$
How to use LU decomposition to solve Ax = b
Prove that there exist a branch
Extension of Sections of Restricted Vector Bundles
Cancellation of Direct Products
Localization of a ring which is not a domain
Basic Probability Question X ~
Proving that $\sum\limits_{i=1}^k i! \ne n^2$ for any $n$
Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$
What is the predual of $L^1$
Use residues to evaluate $\int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x$, where $|a|<1$
How is the value of $\pi$ ( Pi ) actually calculated?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff groups we should have a similar theorem, which would tell us $\{g^{-1}h:g,h\in E\}$ for any subset with positive measure should contain an open neighborhood of the identity.

It seems to me this theorem should be able to tell us a lot about the structure on a topological group, but I cannot really find one example.

Can someone point to some nice applications of Steinhaus in topological groups?

- Is it true that a space-filling curve cannot be injective everywhere?
- isometry $f:X\to X$ is onto if $X$ is compact
- “Truncated” metric equivalence
- Metric is continuous, on the right track?
- Is the boundary of a connected set connected?
- Determine the closure of the set $K=\{\frac{1}{n}\mid n\in\mathbb N\}$ under each of topologies

Thanks!

- Convergence/Sequences/Box Topology
- What's going on with “compact implies sequentially compact”?
- Basic facts about ultrafilters and convergence of a sequence along an ultrafilter
- Unit sphere in $\mathbb{R}^\infty$ is contractible?
- Countable-infinity-to-one function
- A bijective map that is not a homeomorphism
- Normal subgroups of free groups: finitely generated $\implies$ finite index.
- Uniqueness of a continuous extension of a function into a Hausdorff space
- $f$ proper but not universally closed
- Is a maximal open simply connected subset $U$ of a manifold $M$, necessarily dense?

I’m not sure, if this sufficies for an answer..anyhow, I think it’s a neat corollary. I believe this is due to Mackey:

Suppose you have two locally compact, Hausdorff, second countable groups $ G, H $ and a measurable homomorphism $ \phi \colon G \to H $. Then $ \phi $ is actually continuous.

To prove this, start with open neighborhoods $ U,V $ of $ e_{H} $ such that $ VV^{-1} \subset U $. Using the second countability of $ G, H $, we get a countable set $ \{ g_{n} \}_{n \in \mathbb{N}} $ such that $ G = \cup_{n \in \mathbb{N}} g_{n} \phi^{-1}(V) $. Hence there is a $ n_{0} $ with $ m_{G}(g_{n_{0}}\phi^{-1}(V)) > 0 $ and then $ m_{G}(\phi^{-1}(V)) > 0 $ by left invariance. Now as

$ \phi^{-1}(V) \phi^{-1}(V)^{-1} \subset \phi^{-1}(VV^{-1}) \subset \phi^{-1}(U) $

we see by Steinhaus that $ U $ contains an open neighborhood of $ e_{G} $.

- Why is it possible to conclude everything from a false statement?
- Prove that $\int\limits_0^1 x^a(1-x)^{-1}\ln x \,dx = -\sum\limits_{n=1}^\infty \frac{1}{(n+a)^2}$
- What's so special about characteristic 2?
- Scalar Product for Vector Space of Monomial Symmetric Functions
- Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.
- Extension and contraction of ideals in polynomial rings
- Proof of a Gaussian Integral property
- Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3$
- Practical method of calculating primitive roots modulo a prime
- Why don't you need the Axiom of Choice when constructing the “inverse” of an injection?
- Why is completeness theorem true?
- Is it possible for function $f : \mathbb{R} \to \mathbb{R}$ have a maximum at every point in a countable dense subset of its domain?
- If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_n $, prove that $\sum \frac{a_n}{b_n}$ diverges
- Continuous and bounded imply uniform continuity?
- Divisibility rules and congruences