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seen Dec 8 at 21:47

Jul
2
awarded  Curious
Apr
10
awarded  Talkative
Mar
26
accepted Partitions of Unity-Integration on Manifolds
Mar
26
asked Partitions of Unity-Integration on Manifolds
Mar
5
comment When can we use Fubini's Theorem?
Sorry, Reimann integral
Mar
5
comment When can we use Fubini's Theorem?
According to Munkres this might not be true. He says that we should consider the case where a function is integrable but not continuous. Then the function might be integrable over the region but the iterated integral may not because the function may behave badly along a line.
Mar
5
asked When can we use Fubini's Theorem?
Feb
18
awarded  Benefactor
Feb
12
comment Existence of Integral (for a function similar to Thomae's Function)
But if I make the trivial partition, how would things work out. The width would be 1. so Reiman sum would equal 1/q (because there is only 1 Rectangle). I want it to equal 0 (i think 0 is the correct answer).
Feb
12
comment Existence of Integral (for a function similar to Thomae's Function)
Oh. But I thought that in part b) of the question, since the question asks us to compute $\displaystyle \int_{y\in I} f(x,y)$, i just want to fix $x$ and integrate over y.
Feb
12
comment Existence of Integral (for a function similar to Thomae's Function)
I guess I am having difficulty because there are infinitely many rationals in the intervals of a partition and so the function seems to be taking the value 1/q (a fixed number) at infinitely many points. I don't see a way around this.
Feb
12
comment Existence of Integral (for a function similar to Thomae's Function)
The Thomae's function argument works if we integrate over x
Feb
12
comment Existence of Integral (for a function similar to Thomae's Function)
Okay I seem to have an answer but I am slightly confused. Just to confirm, in our case, the $x$ is fixed because the integral is over $y$. So what we want is to consider the upper Riemann sum as $y$ varies - only in the case where $x$ is a fixed rational because otherwise the upper integral is trivially 0. This value fluctuates between 1/q (which is fixed) and 0 depending on y. So we want to construct a partition that in such a scenario, the upper reimann sum is less than epsilon.
Feb
12
comment Existence of Integral (for a function similar to Thomae's Function)
Ok thanks. Part c) is super easy. Part b) the lower integral is obvious and should equal 0. The upper integral defined as infimum(U(f,P)) over all partitions P, where U(f,P) is the upper reimman sum should equal 0. Do I have to use a similar argument of how as partition size decreases, the infimum of U(f,P) will equal 0 because 1/q will approach 0.
Feb
12
accepted Existence of Integral (for a function similar to Thomae's Function)
Feb
12
comment Existence of Integral (for a function similar to Thomae's Function)
Thanks a lot. Is there a way to explicitly show that product measure is 0 just by using the definition of measure 0. The only definition that I am allowed to use is the definition of measure 0. The definition I have is that a set has measure 0 if for every $\epsilon$ you can cover it with countably many rectangles such that the total volume of rectangles is less than $\epsilon$. I am trying to think of how to cover $S_r$ with such rectangles but am not having any success.
Feb
11
comment Existence of Integral (for a function similar to Thomae's Function)
So the only points I need to worry about are (irrational, irrational) and (irrational, rational) and both should have a proof similar to Thomae's function. I didn't understand the part that you said I need to worry about.
Feb
11
comment Existence of Integral (for a function similar to Thomae's Function)
okay. I see. Do you know that how would I establish continuity at points of form (irrational, irrational) and (irrational, rational). Because I would have to show that these two sets of (rational, rational) and (rational, irrational) compose the set of discontinuity.
Feb
11
comment Existence of Integral (for a function similar to Thomae's Function)
The $\delta$-ball would now contain points of form (p/q,irrational) where for every p/q, $q>q_0$.
Feb
11
awarded  Commentator