Timeline for Composed function made Lebesgue integrable?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 30, 2013 at 12:39 | vote | accept | user693 | ||
| Apr 5, 2012 at 11:25 | comment | added | Vincent Beffara | Well, $F$ is assumed to take values in $\mathbb R_+$, and $f$ goes from $\mathbb R_+$ to itself, so $f\circ F$ takes values in $\mathbb R_+$, in other words you do have the assumption that it is finite everywhere (not only a.e. - which would be enough for what you want to do). Some hints then (but if you really want to understand measure theory, you should go through the steps yourself): 1. because $g(x) = \int_0^\infty 1_{y<g(x)} dy$; 2. $\varphi$ is chosen arbitrarily, what you want is $f$; 4. means that there is a unique $f$ satisfying the equality; 5. because $F$ is not in $L^\infty$. | |
| Apr 4, 2012 at 16:04 | comment | added | user693 | This is not homework. I'm still not clear with your answer. Please be more rigorous. Moreover, why $f(F(.))$ is supposed to be a.e. finite? | |
| Apr 4, 2012 at 8:13 | comment | added | Vincent Beffara | For 6., no, this is true as soon as $f\circ F$ is a.e. finite, you do not need integrability. The rest is fairly classical - is it homework? | |
| Apr 3, 2012 at 22:17 | comment | added | user693 | Thanks for the answer. However, it would be great if you can be more precise and rigorous. For instance: 1. why the first equalities follow from Fubini's theorem? 2. why is it guaranteed that there exists a decreasing $\variphi$ such that $\mu(\{f(F(.))=\varphi(u)\})$? 3. what is the "image measure of $\mu$ by $F$"? 4. what do you mean by "this characterizes $f$"? 5. why such an $f$ would start at $0$ and go to $\infty$? 6. why "$\mu(\{f(F(.))>u\})$ always go to $0$ as $u \rightarrow \infty$?" I guess this is true if $f(F(.))$ is integrable, but this is the thesis, not the assumption | |
| Apr 3, 2012 at 21:46 | history | answered | Vincent Beffara | CC BY-SA 3.0 |