Timeline for Interchanging direct sum and direct integral
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 10, 2017 at 15:22 | comment | added | MrUser | It might happen, that two different measurable structures produce different sets of measurable sections, regarding the definition I gave above. So the direct integral would not be unique and would depend on the choice of structure. Then, this equality would be true only for suitable choices. | |
| Aug 9, 2017 at 18:31 | comment | added | MrUser | A section $s$ is measurable with respect to the "measurable structure" (i.e. to the set of measurable orthonormal sections), iff $<s(\omega),e_i(\omega)>$ is measurable for all $i\in\mathbb{N}$. I guess, that was missing. | |
| Aug 9, 2017 at 18:25 | comment | added | Christian Remling | I'm not sure actually I completely understand the definition you summarize, but for me, the elements of a direct integral are measurable (in a suitable sense) functions $s(\omega)$ whose norm, computed in the obvious way, is finite. Then I don't think there will be any problems, though I might be missing something. | |
| Aug 9, 2017 at 18:14 | comment | added | MrUser | Yes, i tried that. This isomorphism induces - obviously - a measurable orthonormal section. But you need to show, that the direct integral is independent of the choice of such a section, so that this isomorphism induces the same direct integral, don't you? | |
| Aug 9, 2017 at 17:33 | comment | added | Christian Remling | Also, the statement has to be true. If it isn't, then this just means that one has to revise definitions. | |
| Aug 9, 2017 at 17:33 | comment | added | Christian Remling | Equality of course must be interpreted as "there is an obvious isomorphism," and I think the only map that comes to mind is the one that sends $s(\omega)=(s_i)_{i\in\mathbb N}(\omega)$ to $(s_i(\omega))_i$ (in other words, it doesn't do anything other than slightly reinterpret). Have you tried if this works? | |
| Aug 9, 2017 at 9:28 | history | edited | Ben McKay | CC BY-SA 3.0 |
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| Aug 9, 2017 at 9:14 | history | edited | MrUser | CC BY-SA 3.0 |
added 62 characters in body
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| Aug 9, 2017 at 9:13 | review | First posts | |||
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| Aug 9, 2017 at 9:09 | history | asked | MrUser | CC BY-SA 3.0 |