Timeline for Measure theory treatment geared toward the Riesz representation theorem
Current License: CC BY-SA 3.0
12 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Aug 22, 2012 at 0:41 | comment | added | Yemon Choi | @LeBlanc: I meant that the RRT is more general for $C_c(X)$ because it characterizes functionals on a smaller algebra; it is not a priori clear that the functional on $C_c(X)$ extend to functionals on $C_0(X)$ without further assumptions on $X$. | |
| Aug 21, 2012 at 23:03 | history | edited | LeBlanc | CC BY-SA 3.0 |
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| Aug 21, 2012 at 22:55 | comment | added | LeBlanc | @Yemon, How can that be? For example, if $X=\mathbb R$, $f(x)=e^{-x^2}$ is an element of $C_0(X)$ but not of $C_c(X)$. | |
| Aug 21, 2012 at 10:47 | history | edited | LeBlanc | CC BY-SA 3.0 |
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| Aug 21, 2012 at 10:43 | comment | added | Igor Khavkine | I see. In my understanding, treating $C_0(X)$ is the same as treating $C(\hat{X})$, where $\hat{X}$ is the one-point compactification. So it seems that Rudin only goes as far as treating $C(X)$ for compact $X$. | |
| Aug 21, 2012 at 10:39 | comment | added | Yemon Choi | I thought C_c(X) is more general than C_0(X), since it's a subalgebra? | |
| Aug 21, 2012 at 10:13 | history | edited | LeBlanc | CC BY-SA 3.0 |
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| Aug 21, 2012 at 10:08 | comment | added | LeBlanc | @Igor Please see edit. Rudin does treat a more general case. | |
| Aug 21, 2012 at 10:08 | history | edited | LeBlanc | CC BY-SA 3.0 |
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| Aug 21, 2012 at 9:58 | comment | added | Igor Khavkine | Thanks for looking checking! BTW, I just looked up Halmos, and he also seems to treat only the $C_K(X)$ case. | |
| Aug 21, 2012 at 9:51 | history | answered | LeBlanc | CC BY-SA 3.0 |