Timeline for Integration under functional sign
Current License: CC BY-SA 3.0
3 events
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| May 8, 2012 at 20:35 | comment | added | Phil Isett | (It looks like there is a coding error in the comment.) I think I'm starting to see the relationship between these two arguments. On the one hand, your argument appeals to Fubini to get started; the argument that I suggested actually avoids Fubini's theorem by taking Riemann sums. I actually thought that you couldn't get away without this move (for the reason in my post), but it looks like you can just get by with the continuity in the $y$ variable? Thoughts? | |
| May 8, 2012 at 16:23 | comment | added | Liviu Nicolaescu | @ Phil The collection of functions $\Phi:=(\; f(-,y)\;)_{y\in \Omega}$ is bounded in $\mathscr{E}(\Omega)$ due to (C). Set $$s(\varepsilon):=\sup_{y\in \Omega} |(L_{\varepsilon, x}-L_x)f(x,y)|$$ The uniform boundedness principle implies $s(\varepsilon)\to 0$. Hence $$\left|\;\int_\Omega(L_{\varepsilon, x}-L_x)f(x,y)d\mu(y)\;\right| \leq \int_Omega s(\varepsilon) d\mu(y)=s(\varepsilon)\to 0. $$ | |
| May 8, 2012 at 16:03 | history | answered | Phil Isett | CC BY-SA 3.0 |