Timeline for The kernel of all invariant means
Current License: CC BY-SA 4.0
11 events
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| Mar 7, 2022 at 14:28 | comment | added | ARG | @Lspice oh! cool, thanks for the latex-info! As for the math: it is not trivial to me! (that's most of the content of the answer). (1) any element of the kernel satisfies (*) (the equation with $\displaystyle \lim_{i \to \infty} \sup_{x \in G} ... =0$) (2) so do the $f_i$ (because they also have 0 mean. (3) $f-f_i \in S$ (it's a convex combination of some $f - \lambda_g f$) (4) because of (2), the $f-f_i$ tend to $f$ (for this you have to interpret (2) as saying that the $f_i \to 0$ in norm). (5) this is how you show $K$ is contained in the closure of $S$. | |
| Mar 7, 2022 at 14:24 | comment | added | LSpice | Sorry for the foolish question, but why is it obvious that the trivial answer gives the full kernel? It is obvious that it is contained in the kernel, but why is it obvious that there are enough invariant means that nothing outside the span you indicate is annihilated by all of them? | |
| Mar 7, 2022 at 14:22 | comment | added | LSpice |
@ARG, \displaystyle doesn't convert $\cap$ \cap to $\bigcap$ \bigcap; see $\cap\bigcap\displaystyle\cap\bigcap$ \cap\bigcap\displaystyle\cap\bigcap. You always need to use it manually when appropriate.
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| Mar 7, 2022 at 13:17 | history | edited | YCor | CC BY-SA 4.0 |
\cap->\bigcap
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| Mar 7, 2022 at 13:15 | comment | added | ARG | ah, sorry... I usually use \displaystyle (which normally turns the $\cap$ into $\bigcap$ automatically) but forgot to do it here | |
| Mar 7, 2022 at 13:06 | comment | added | YCor | @ARG it's fine; just prefer $\bigcap$ (bigcap) for multiple intersections to $\cap$ (cap) which is only for intersections as binary operator. | |
| Mar 7, 2022 at 13:01 | comment | added | ARG | I made quite a big edit, please feel free to roll back if I messed it up. Reading this proof after a few year, the crucial point that elements of $K$ are limits of elements of $S$ was hard to understand. | |
| Mar 7, 2022 at 12:59 | history | edited | ARG | CC BY-SA 4.0 |
tried to make the step that any element of $K$ is a limit of elements of $S$ more explicit
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| Jun 7, 2017 at 16:50 | vote | accept | ARG | ||
| S Jun 5, 2017 at 12:56 | history | answered | YCor | CC BY-SA 3.0 | |
| S Jun 5, 2017 at 12:56 | history | made wiki | Post Made Community Wiki by YCor |