Timeline for Post composition of integral
Current License: CC BY-SA 3.0
Post Revisions
14 events
| when toggle format | what | by | license | comment | |
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| Oct 6, 2015 at 17:44 | history | edited | Johannes Hahn | CC BY-SA 3.0 |
Fixed notation errors. Hopefully everyone understands now who is integrated over whom.
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| Oct 6, 2015 at 15:14 | review | Close votes | |||
| Oct 10, 2015 at 14:15 | |||||
| Oct 6, 2015 at 14:12 | comment | added | Giovanni De Gaetano | Thanks, now I get it. But I still think that your defining formula should read $$(\forall x^* \in E^*)\quad x^*(f) = \int_\Omega x^*(f(\omega)) d\mu(\omega).$$ Then $\int_E f d \mu \in E^{**} \simeq E$ is the mapping $(\int_E f d \mu)(x^*) = x^*(f)$. | |
| Oct 6, 2015 at 14:07 | comment | added | AB_IM | It's okay, I clarified the notation further. | |
| Oct 6, 2015 at 14:05 | history | edited | AB_IM | CC BY-SA 3.0 |
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| Oct 6, 2015 at 14:04 | answer | added | AB_IM | timeline score: 1 | |
| Oct 6, 2015 at 14:02 | comment | added | Giovanni De Gaetano | Do you mean $x^*(f)$ instead of $x^*(e)$? And I still don't get what is denoted by $\int_E f d \mu$, because the expression above seems to be dependent on the choice of $x^* \in E$. Moreover, $x$ is an element of $\Omega$ and $x^*$ of $E$? I apologize for the annoying intrusion... | |
| Oct 6, 2015 at 13:53 | comment | added | AB_IM | I edited my post to clarify my notation. | |
| Oct 6, 2015 at 13:52 | history | edited | AB_IM | CC BY-SA 3.0 |
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| Oct 6, 2015 at 13:46 | history | edited | AB_IM | CC BY-SA 3.0 |
added 85 characters in body
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| Oct 6, 2015 at 13:40 | comment | added | Liviu Nicolaescu | I meant, what do you mean by the integral of $f$ over $E$ when $f$ is defined on $\Omega$? | |
| Oct 6, 2015 at 13:11 | comment | added | AB_IM | Its the unique $e\in E$ solving: $(x^* \in E^*)\, x^*(e) = \int x^*(f) $ (where the right is the Lebesgue integral). | |
| Oct 6, 2015 at 13:01 | comment | added | Liviu Nicolaescu | Can you define $\int_E f$ where $f:\Omega\to E$? | |
| Oct 6, 2015 at 12:48 | history | asked | AB_IM | CC BY-SA 3.0 |