Understanding the ##\epsilon## definition of this integral

[Adesh]

TL;DR Summary

A function $f$ which is bounded on $[a,b]$ is integrable on $[a,b]$ if and only if for any $\epsilon~~\gt 0$ we can find a partition P of $[a,b]$ such that $$U(f,P) - L(f,P) \lt \epsilon$$

Integrals are defined with the help of upper and lower sums, and more number of points in a partition of a given interval (on which we are integrating) ensure a lower upper sum and a higher lower sum. Keeping in mind these two things, I find the following definition easy to digest

A function $f$ which is bounded on $[a,b]$ is integrable on $[a,b]$ if and only if
$$sup \{L(f,P) : \text{P is the partition of [a,b]}\} = inf\{U(f,P) : \text{P is a partition of [a,b]}\}$$
and integral of $f$ on $[a,b]$ is that same common number.However, it is said that the above definition is not handy and hard to use (I don't know why, I find it better) so we have another definition (derivable from the above one) which says

A function $f$ which is bounded on $[a,b]$ is integrable on $[a,b]$ if and only if for any $\epsilon~~\gt 0$ we can find a partition P of $[a,b]$ such that $$U(f,P) - L(f,P) \lt \epsilon$$

I'm having so many doubts in the above definition, what does it exactly mean to say "we can find a partition P such that ...", how can we ensure that we can find a partition P? And also this second definition doesn't say anything about what the integral will be. Please explain it to me how YOU PERSONALLY understand that $\epsilon$ thing.

 

[FactChecker]

The definition you like allows you to say that there is a sequence of partitions, $P_{sup,i}$ that gets as close to the supremum as you want. Likewise, there is a sequence of partitions, $P_{inf,i}$ that gets as close to the infimum as you want. If there is a partition that gives the exact value, that is also ok. Suppose we specify an $\epsilon \gt 0$. Pick an $P_{sup,i}$ within $\epsilon/2$ of the supremum and a $P_{inf,j}$ within $\epsilon/2$ of the infimum. Combine these two partitions into a finer partition, $P^*$. It satisfies the requirements of the second definition.

 

[Delta2]

The first definition essentially tell us that as the partition P gets finer and finer, and $U(f,P)$ gets smaller and smaller while $L(f,P)$ gets bigger and bigger , the imposing of the equality of the (sup=inf )essentially means that the difference $U(f,P)-L(f,P)$ gets as small as we want (smaller than any $\epsilon>0$) as long as we choose a partition P that is "fine enough" (has small enough norm or mesh). But that's essentially what's the second definition tell us.

The second definition reminds me in some way the epsilon-delta definition of the limit in the sense that U(f,P) converges to L(f,P) (or vice versa), where the role of delta is played by the norm (or mesh) of the partition (the number that tell us how fine the partition is).

 

[member 587159]

The second definition reminds me in some way the epsilon-delta definition of the limit in the sense that U(f,P) converges to L(f,P) (or vice versa), where the role of delta is played by the norm (or mesh) of the partition (the number that tell us how fine the partition is).

Because it in some way is a limit. Every limit you encounter is a special case of limit of a net.

 

[Adesh]

The definition you like allows you to say that there is a sequence of partitions, $P_{sup,i}$ that gets as close to the supremum as you want. Likewise, there is a sequence of partitions, $P_{inf,i}$ that gets as close to the infimum as you want. If there is a partition that gives the exact value, that is also ok. Suppose we specify an $\epsilon \gt 0$. Pick an $P_{sup,i}$ within $\epsilon/2$ of the supremum and a $P_{inf,j}$ within $\epsilon/2$ of the infimum. Combine these two partitions into a finer partition, $P^*$. It satisfies the requirements of the second definition.

Does the partition gets close to the supremum? I missed something and unable to understand your nice reply. Please help me a little more.

 

[Adesh]

The first definition essentially tell us that as the partition P gets finer and finer, and $U(f,P)$ gets smaller and smaller while $L(f,P)$ gets bigger and bigger , the imposing of the equality of the (sup=inf )essentially means that the difference $U(f,P)-L(f,P)$ gets as small as we want (smaller than any $\epsilon>0$) as long as we choose a partition P that is "fine enough" (has small enough norm or mesh). But that's essentially what's the second definition tell us.

The second definition reminds me in some way the epsilon-delta definition of the limit in the sense that U(f,P) converges to L(f,P) (or vice versa), where the role of delta is played by the norm (or mesh) of the partition (the number that tell us how fine the partition is).

What causes me a problem is the English sentence “we can always find a partition P such that the following condition is true”. Would you please elabaorate on that?

I have understood that $$U(f,P) - L(f,P) \lt \epsilon$$ means we can make upper sum as close to lower sum as we want.

 

[member 587159]

I understand this intuitively in the following way: The difference $U(f,P)- L(f,P)$ can be made arbitrary small. This means that the difference between the upper sum and the lower sum get arbitrary small, in fact so small that there is a unique number that can be sandwiched behind them. This will be the integral.

Try to prove the two definitions you gave are equivalent. The intuition of what happens should come naturally.

 

[Adesh]

in fact so small that there is a unique number that can be sandwiched behind them.

Means they get so close that the elementary Theorem “between any two numbers there lies infinitely many numbers” gets violated and there left just one number between them?

 

[member 587159]

No, it does not get violated, because 'between any two numbers' does not apply here. $U(f,P), L(f,P)$ vary when $\Vert P \Vert \to 0$, so the two numbers you are talking about are not fixed.

 

[Adesh]

This means that the difference between the upper sum and the lower sum get arbitrary small, in fact so small that there is a unique number that can be sandwiched behind them.

So, a stage would come where there is just one number between $U(f,P)$ and $L(f,P)$ ? Doesn’t that really mean that only one number will lie between two numbers (after some stage of “refining”) ?

Well, we can find a partition such that difference between upper and lower sum is very small, and hence we can make as close to each other as we desire. But “between any two numbers, no matter how close they are, there always lies infinitely many numbers between them”.

 

[member 587159]

So, a stage would come where there is just one number between $U(f,P)$ and $L(f,P)$ ? Doesn’t that really mean that only one number will lie between two numbers (after some stage of “refining”) ?

Well, we can find a partition such that difference between upper and lower sum is very small, and hence we can make as close to each other as we desire. But “between any two numbers, no matter how close they are, there always lies infinitely many numbers between them”.

It remains a limiting process. Compare with the following:

$0$ is the unique number such that $-1/n <0 <1/n$ for all $n\geq 1$.

 

[Delta2]

Well I believe you already know that as the partition P becomes finer and finer (the norm or mesh of the partition becoming smaller and smaller) , then U(f,P) becomes smaller and smaller and comes closer to its infimum (while L(f,P) becomes bigger and bigger and comes closer to its supremum). To make this abit more rigorous we have to define a sequence of Partitions $P_n$ such that the sequence of the mesh of these partitions, $\lambda_n$ has limit zero. Then we can write down the following
Given that $\lim\lambda_n=0$ we ll have that $$\lim U(f,P_n)=inf(U(f,P)|P partition)$$ and $$\lim L(f,P_n)=sup(L(f,P)|P partition)$$
if furthermore it holds that $$inf(U(f,P))=sup (L(f,P))=L$$ then you can prove (using the $\epsilon$ definition of the limit of a sequence , for the sequences $U(f,P_n)$ and $L(f,P_n)$) that for every $\epsilon>0$ there exists $n_0$ such that for all partitions $P_n,n\geq n_0$ we ll have
$$U(f,P_n)-L<\frac{\epsilon}{2}$$
$$L-L(f,P_n)<\frac{\epsilon}{2}$$
And if you add the two inequalities above you get the desired result.

 

[FactChecker]

Does the partition gets close to the supremum? I missed something and unable to understand your nice reply. Please help me a little more.

EDIT CORRECTION: I had the U and L swapped. I have corrected them.
What is your definition of the supremum? You can find some partition with L as close as you like to the supremum, say $P_{sup,\epsilon/2}$ within $\epsilon/2$. Any refinement of $P_{sup,\epsilon/2}$ will have L at least as close, maybe closer. Similarly for the infimum. So given an arbitrarily small $\epsilon \gt 0$, you can find a pair, $P_{sup,\epsilon/2}$ and $P_{inf,\epsilon/2}$ where the corresponding L and U summations are within $\epsilon/2 + \epsilon/2 = \epsilon$. You can merge the two partitions into $P^*$, which is a refinement of both partitions and so the respective L and U summations for $P^*$ are at least as close to both the supremum and infimum as $\epsilon/2$. That should tell you that $P^*$ satisfies the requirement of the second definition.

 

[Mark44]

Does the partition gets close to the supremum?

No, they are two different things. The partition is the set of points at which the function f is evaluated to get either an upper sum or a lower sum. The supremum is the upper bound on the lower sums.
Geometrically, the upper and lower sums are the areas of rectangles that, respectively, overestimate the value of the integral and underestimate the value of the integral.

 

[Adesh]

You can find some partition with U as close as you like to the supremum, say Psup,ϵ/2Psup,ϵ/2P_{sup,\epsilon/2} within ϵ/2ϵ/2\epsilon/2.

Is that a typo (beacuse we want U to attain infinimum) or did you do that for explanatory purpose?

 

[Adesh]

there exists n0n0n_0 such that for all partitions Pn,n≥n0Pn,n≥n0P_n,n\geq n_0 we ll have

U(f,Pn)−L<ϵ2U(f,Pn)−L<ϵ2​

U(f,P_n)-LL−L(f,Pn)<ϵ2L−L(f,Pn)<ϵ2L-L(f,P_n)

I want to that we just assured “for some $P_n$, $n \gt n_0$” we had $$ U(f,P_n) -L \lt \epsilon/2 \\
L- L(f,P_n) \lt \epsilon/2 $$ but I want to ask if same $n_0$ going to work for both of them? Or did we take our $n_0$ greater than both individual $n’_0$ and $n’’_0$ ( $n’_0$ is such that for any $P_n$, $n \gt n’_0$, such that $U(f,P_n)$ is within $\epsilon/2$ of L, and similarly for $n’’_0$ is such that for any $P_n$, $n \gt n’’_0$ such that $L(f,P_n)$ is within $\epsilon/2$ of L)

 

[Adesh]

(Can somebody please tell me why I don’t get mathjax coded thing when I quote someone’s mathematical equations/expressions)

 

[Delta2]

I want to that we just assured “for some $P_n$, $n \gt n_0$” we had $$ U(f,P_n) -L \lt \epsilon/2 \\
L- L(f,P_n) \lt \epsilon/2 $$ but I want to ask if same $n_0$ going to work for both of them? Or did we take our $n_0$ greater than both individual $n’_0$ and $n’’_0$ ( $n’_0$ is such that for any $P_n$, $n \gt n’_0$, such that $U(f,P_n)$ is within $\epsilon/2$ of L, and similarly for $n’’_0$ is such that for any $P_n$, $n \gt n’’_0$ such that $L(f,P_n)$ is within $\epsilon/2$ of L)

You are right here , they are actually $n_0',n_0''$ and we have to take $n_0=max({n_0',n_0''})$.

 

[Delta2]

(Can somebody please tell me why I don’t get mathjax coded thing when I quote someone’s mathematical equations/expressions)

Forum recently upgraded to Mathjax 3 (from 2) but it turned out that 3 had many bugs, main bug being that the preview function didnt render latex commands. So they rolled back to Mathjax 2. Maybe your issue relates to that.

 

[FactChecker]

Is that a typo (beacuse we want U to attain infinimum) or did you do that for explanatory purpose?

Good catch. Just a mistake because I wasn't thinking; not a typo. I'll go back and correct it. Thanks.

 

[Adesh]

I just want to thank @FactChecker and @Delta2 for explaining me. Their understanding of the subject is very clear and communicable. I don’t find Spivak as clear as they are, although Spivak’s book is very famous and people often don’t complain about it so I have to admit (in front of world) that problem lies within me. Thank you again both of you for explaining this thing so clearly to me. (Sam Harris said that “If you want to thank somebody then from next time try to mean it” :-)

 

[member 587159]

I just want to thank @FactChecker and @Delta2 for explaining me. Their understanding of the subject is very clear and communicable. I don’t find Spivak as clear as they are, although Spivak’s book is very famous and people often don’t complain about it so I have to admit (in front of world) that problem lies within me. Thank you again both of you for explaining this thing so clearly to me. (Sam Harris said that “If you want to thank somebody then from next time try to mean it” :-)

The best exposition of the Riemann integral I have seen is in the book "The real numbers and real analysis" by Ethan Block, so if something is unclear in Spivak you might want to have a look there.

 

[Adesh]

The best exposition of the Riemann integral I have seen is in the book "The real numbers and real analysis" by Ethan Block, so if something is unclear in Spivak you might want to have a look there.

Thank you so much for that suggestion, I will surely have a look on it.

 

[FactChecker]

I don’t find Spivak as clear as they are, although Spivak’s book is very famous and people often don’t complain about it so I have to admit (in front of world) that problem lies within me.

It's not unusual at all for a textbook to be too terse for one to easily understand. It happens all the time to all of us. That is where it is a great advantage to be able to discuss it with others, either with other students or in a forum like this.