A challenging question on Riemann integration ( Reference- Real Analysis/A.J.White)

Question

If $f:[a,b]→R$ is Riemann integrable over [a,b] and $f≥0$, then given $ϵ>0$ there exists $δ>0$ such that if $P=(y_1,...,y_m)$ (which is a partition of [a.b]) and max{$y_{s+1}$-$y_{s}$:s=1,...,m-1}<$δ$, then $U(P,f)-\int_{a}^{b}f<ϵ$. $$$$I would much appreciate if someone could help me out with this question. Thanks

http://math.stackexchange.com/questions/457878/to-prove-the-equivalence-definition-of-riemann-integral — Prahlad Vaidyanathan, Nov 03 '13 at 05:32
"A detailed canonical answer is required to address all the concerns." Which concerns are not addressed by the link? Note that the question itself is or is not empty, depending on the definition of Riemann integrability one uses. Please explain. // Upvoters: why the upvotes? — Did, Dec 16 '13 at 10:33
This is a good question. Hoping to see a simple answer :) +1 from me — , Dec 16 '13 at 10:42
@user81883 Thanks for answering my query. Why "a good question"? Did you see that there ALREADY IS an answer? — Did, Dec 16 '13 at 11:04
Yes I saw it but I'm sure the OP put a bounty up since he couldn't understand it. From the level of knowledge I have I also could not understand much of it. Maybe a clarification of the link would do :) @Did — , Dec 16 '13 at 11:11
@Did I don't think I found the answer I am looking for from the link although it was quite useful. I put up the for anyone to give me a good answer. Or maybe you could explain the answer in link for me . — Heisenberg, Dec 16 '13 at 11:17
Nobody can answer your question without knowing what your definition of Riemann integrable is. You will also need to tell us what $U(P,f)$ is: a Riemann sum? the "upper Darboux sum"? something entirely different? — mrf, Dec 16 '13 at 11:29
@mrf U(P,f) is the upper sum. Riemann integrable meaning for each ϵ>0 there exists a partition P such that U(P,f)-L(P,f)<ϵ — Heisenberg, Dec 16 '13 at 11:43
@user81883 "Maybe a clarification of the link would do"... Sure, IN WHICH direction? To explain WHICH point? Do you understand nothing, a part of it, everything but a step, none of the above? "I do not understand, please add clarification" is wonderfully (?) UNinformative, don't you think? — Did, Dec 16 '13 at 12:20
@RajindaWickrama "maybe you could explain the answer in link for me"... Good idea, unfortunately I CANNOT DO THAT as long as you stay silent about what you are looking for. This is the problem with asking "naked" questions, with no context, no explanation, no indication of the required level, of the things you master and the ones you don't, and so on. — Did, Dec 16 '13 at 12:23
@RajindaWickrama Now that you added a definition, I do not understand: you know that U(P,f)-L(P,f)<ϵ and that L(P,f)<I<U(P,f) and you want to show that U(P,f)-I<ϵ, right? Seriously... — Did, Dec 16 '13 at 12:24
@Did I figured out the answer my self which is much different from what is in the link :) This was directly off a text book as I mentioned. Questioning me about the authors question is pointless I asked this question because I don't know the answer to it. I included every word in the question that I took off the book. If any more explanation was required I will have to ask the author or assume it myself and say it which may not be accurate. But finally I found the answer myself which I will upload to the site later. — Heisenberg, Dec 16 '13 at 12:33
@RajindaWickrama I am NOT "questioning you about the authors question", I am giving you precise indications to reach a proof (and I did it as soon as it was possible to do so, that is, when you accepted to complete your question). Sorry but your attitude is most unconstructive. — Did, Dec 16 '13 at 12:36
@Did: This is about equivalence of two definition of Riemann integration. Let $U(P, f), L(P, f), S(P, f)$ denote upper sum, lower sum, Riemann sum for a tagged partition respectively. Def 1) $f$ is R-integrable over $[a, b]$ if for any $\epsilon > 0$ there is a partition $P$ of $[a, b]$ such that $U(P, f) - L(P, f) < \epsilon$. Def 2) $f$ is R-integrable over $[a, b]$ if there is a number $I$ such that for any $\epsilon > 0$ there is a $\delta > 0$ for which $|S(P, f) - I| < \epsilon$ whenever the length of largest subinterval in $P$ is less than $\delta$. — Paramanand Singh, Dec 17 '13 at 16:14
Adding to my prev comment, the usual definition is Def 2) and what I have mentioned as Def 1) is a standard result to be derived further. Note that Def 1) does not say anything about value of integral, but def 2) does so. Hence def 2) looks more proper as a definition whereas def 1) is more like a criterion for integrability. — Paramanand Singh, Dec 17 '13 at 16:16
Also the current question uses the obvious result $L(P, f) \leq S(P, f) \leq U(P, f)$ and then combines it with def 2) which gives rise to $U(P, f) - I < \epsilon$ for all $P$ with mesh less than $\delta$ — Paramanand Singh, Dec 17 '13 at 16:23
@ParamanandSingh Compare with my answer. (You know what, I KNOW how to solve this. :-) Funny to be lectured on Riemann integral though... made me feel... well, an odd feeling anyway.) — Did, Dec 17 '13 at 17:03
@Did: First of all very sorry, if you feel odd (or offended). My point was not to lecture on Riemann stuff because frankly speaking I am much novice enough to lecture most of the people on MSE. I was bit confused by your last comment "Now that you added a definition, I do not understand". Also I wanted to point out that the main point of question was to make a transition from "one partition P" to "all partitions with mesh less than some $\delta$". Sorry once again if my comment has hurt you. — Paramanand Singh, Dec 17 '13 at 17:46

score 1 · Answer 1 · answered Dec 16 '13 at 12:39

1

Complete answer: $[L\leqslant I\leqslant U\land U-L\leqslant\varepsilon]\implies U-I\leqslant\varepsilon$.

answered Dec 16 '13 at 12:39

Did

279,727

what is U^U? I don't know this notation – Dec 16 '13 at 12:43
1

$\land$ is the logical AND. – Did Dec 16 '13 at 12:44
Meaning L<=U−L? – Dec 16 '13 at 12:47
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@user81883 ??? $[P\land Q]$ means that both $P$ and $Q$ hold, here both $L\leqslant I\leqslant U$ and $U-L\leqslant\varepsilon$. – Did Dec 16 '13 at 13:02
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So of course it is better to write "and" than to write "$\wedge$". – GEdgar Dec 16 '13 at 15:54
@GEdgar Who says so? – Did Dec 16 '13 at 18:18

score -1 · Accepted Answer · answered Dec 17 '13 at 14:14

-1

Here is the answer I figured out and I am putting it here.

Let ε>0 be arbitrary. Then there exists a partition $P_1$={$x_0$,...,$x_p$} in $[a,b]$ such that $U(P_1,f)<\int_{a}^{b}f+ϵ/2$

Choose $δ=ϵ/4K(p-1)$ where K is the upper bound of $f(x)$.

Let P be any partition of $[a,b]$ with $||P||<δ$.

Let $P'$ be the union of $P$ and $P_1$ .

$P'$ has at most $(p-1)$ points that $P$. $$U(P,f)-2K(p-1)δ⩽U(P',f)⩽U(P_1,f)<\int_{a}^{b}f+ϵ/2$$ Thus,

$$ U(P,f)<\int_{a}^{b}f+ϵ$$ for any partition with $||P||<δ$.

answered Dec 17 '13 at 14:14

Heisenberg

3,227

Wow!! I hope this is right. It looks right to me. Can't believe I missed something so simple all this time. Best answer so far! +1 – Dec 17 '13 at 14:20
@user81883 Thanks alot it should be correct but open for any criticism – Heisenberg Dec 17 '13 at 14:21
1

Your answer is correct, however why do you need the condition $f \geq 0$ in your question. Your proof as well as the question is correct without this assumption. – Paramanand Singh Dec 17 '13 at 16:06

A challenging question on Riemann integration ( Reference- Real Analysis/A.J.White)

2 Answers2