Timeline for Using the Stone-Weierstrass theorem to solve an integral limit
Current License: CC BY-SA 4.0
13 events
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| Jan 6 at 21:59 | history | edited | Christian Remling | CC BY-SA 4.0 |
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| Jan 6 at 21:37 | comment | added | Christian Remling | @MartinVäth: That neatly confirms my theory that this (= substitution rule in the setting of general image measures) is the most neglected topic in measure theory/integration textbooks. By the way, the proof is by chasing definitions and almost does itself. | |
| Jan 6 at 21:34 | comment | added | Martin Väth | Thanks: So far, I was only aware that integration by parts and substitution rule hold for absolutely continuous (substitution) functions. Thanks for the link to the general case. | |
| Jan 6 at 21:31 | comment | added | Christian Remling | @MartinVäth: Or even integration by parts: $\int_0^1 f\, df = f^2\bigr|_0^1-\int_0^1 f\, df$: en.wikipedia.org/wiki/… | |
| Jan 6 at 21:29 | comment | added | Christian Remling | @MartinVäth: This is immediate from a sufficiently general version of the substitution rule since $\mu$ is the image measure of Lebesgue measure under $f:[0,1]\to[0,1]$, so $\int_0^1 f\, d\mu = \int_0^1 x\, dx$. See here: en.wikipedia.org/wiki/… | |
| Jan 6 at 21:23 | comment | added | Martin Väth | Nice argument, but the problem does not seem to be solved completely: For absolutely continuous $f$ with $f(0)=0$ and $f(1)=1$, it seems to be rather straightforward to see that the latter integral is $1/2$. Can this be shown for the general case as well? (In fact, it seems plausible for the devli's staircae, but I do not see a rigorous proof.) | |
| Jan 6 at 18:55 | history | edited | Christian Remling | CC BY-SA 4.0 |
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| Jan 6 at 16:44 | comment | added | Christian Remling | If Lebesgue integrals are off limits (as indicated in the OP), this should also be fine when the integrals $\int d\mu\ldots$ are interpreted as Riemann-Stieltjes integrals (though this seems about as meaningful as being asked to play a piano piece, touching the keys only with one's nose). | |
| Jan 6 at 16:40 | history | edited | Christian Remling | CC BY-SA 4.0 |
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| Jan 6 at 14:51 | history | undeleted | Christian Remling | ||
| Jan 6 at 14:51 | history | edited | Christian Remling | CC BY-SA 4.0 |
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| Jan 6 at 14:44 | history | deleted | Christian Remling | via Vote | |
| Jan 6 at 14:41 | history | answered | Christian Remling | CC BY-SA 4.0 |