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Nov 6, 2007, 11:55:23 PM11/6/07

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i've just learned that the definite integral of f is equal to the

limit of the riemann midpoint sum if f is continuous in the interval.

i'm curious -- can that limit exist for a function that's not riemann

integrable? if so, what would such a function be? if not, why not?

limit of the riemann midpoint sum if f is continuous in the interval.

i'm curious -- can that limit exist for a function that's not riemann

integrable? if so, what would such a function be? if not, why not?

thanks,

ame

Nov 7, 2007, 2:18:36 AM11/7/07

to

Wlog. your interval is [0,1].

If you take the limit only for equidistant subintervals (or similarly

for any fixed sequence of subintervals), i.e. your n'th term of the

limit is

sum_{0<=k<n} f( (k + 1/2)/n )

then the counterexample is simple:

This limit is already 0 if f(q) is zero for all rationals q.

If f(x)=1 for all irrational x then f is not Riemann integrable

although the above limit exists. Also, the limit differs from the

Lebesgue integral, which would be 1.

hagman

Nov 7, 2007, 2:56:06 AM11/7/07

to

ame <rae...@gmail.com> writes:

Yes it can exist. Let's say your interval is [0,1]. Your Riemann midpoint

sums all involve evaluating f at rational numbers. Consider a function

that is 0 on the rationals (so all the Riemann midpoint sums are 0), but

does something wild and crazy on the irrationals...

--

Robert Israel isr...@math.MyUniversitysInitials.ca

Department of Mathematics http://www.math.ubc.ca/~israel

University of British Columbia Vancouver, BC, Canada

Nov 7, 2007, 3:45:16 AM11/7/07

to

On 2007-11-07 02:56:06 -0500, Robert Israel

<isr...@math.MyUniversitysInitials.ca> said:

<isr...@math.MyUniversitysInitials.ca> said:

> ame <rae...@gmail.com> writes:

>

>> i've just learned that the definite integral of f is equal to the

>> limit of the riemann midpoint sum if f is continuous in the interval.

>> i'm curious -- can that limit exist for a function that's not riemann

>> integrable? if so, what would such a function be? if not, why not?

>

> Yes it can exist. Let's say your interval is [0,1]. Your Riemann midpoint

> sums all involve evaluating f at rational numbers. Consider a function

> that is 0 on the rationals (so all the Riemann midpoint sums are 0), but

> does something wild and crazy on the irrationals...

You seem to be considering only nets of partitions with points in the

rationals only. In that case, all midpoints are rationals also.

However, what if you consider also nets of partitioners with points in

all of reals too?

For a partition with points in the irrationals, the midpoints need not

be rationals. So, the limit of riemann midpoint sum will not converge

in your example. That is, I can always find a finer partition with

irrational points such that the midpoints are also irrational.

Did I misunderstood something here?

--

-kira

Nov 7, 2007, 10:41:19 AM11/7/07

to

Kira Yamato wrote:

> You seem to be considering only nets of partitions

> with points in the rationals only. In that case,

> all midpoints are rationals also. However, what if

> you consider also nets of partitioners with points

> in all of reals too?

>

> For a partition with points in the irrationals, the

> midpoints need not be rationals. So, the limit of

> riemann midpoint sum will not converge in your example.

> That is, I can always find a finer partition with

> irrational points such that the midpoints are also

> irrational.

I posted some remarks about this issue, and the issue

of which point is selected in the various intervals,

in these December 10, 2002 sci.math posts:

http://groups.google.com/group/sci.math/msg/07d2e1f77af46765

http://groups.google.com/group/sci.math/msg/c8fd54351693a760

I thought I had also, at some later time, posted citations

to the papers I alluded to in the first post, but I can't

find to post now, so maybe I never made it. I'll post some

references tomorrow. (I have to dig them up when I get

home tonight.)

Dave L. Renfro

Nov 7, 2007, 3:20:05 PM11/7/07

to

Dave L. Renfro wrote (in part):

> I thought I had also, at some later time, posted citations

> to the papers I alluded to in the first post, but I can't

> find to post now, so maybe I never made it. I'll post some

> references tomorrow. (I have to dig them up when I get

> home tonight.)

I just remembered that it wasn't a post, but rather it was

an e-mail to someone whose preliminary real analysis text

manuscript I was giving editing feedback on. The e-mail

is below, with some slight changes and omissions that have

to do with specifics about the person I was writing to.

----------------------------

I believe Cauchy proved the left endpoint integral exists

for continuous functions on compact intervals (modulo his

uniform continuity omission) and he also proved that all

Riemann sums for such a function converge to the same result.

I think his definition was the left endpoint version and its

agreement with Riemann's formulation was a theorem, but I'm

not entirely sure about this. In any event, the Riemann

version of the integral had been around much earlier than

Riemann lived, and I think this is fairly well known. I

believe that, even back in Newton's time, mathematicians dealt

with general "Riemann sums" when computing areas and other

things (i.e. the location of the evaluation point in

each interval was varied as needed for the problem at

hand). What Riemann mainly did was to put the focus

on the collection of functions that are integrable

according to some notion of integrability, rather than

(as Cauchy did) defining a notion of integrability only

to be able to rigorously prove certain desired integrability

properties.

Gillespie [1] proved that any bounded function on a compact

interval is "left endpoint integrable" if and only if it is

Riemann integrable.

Kristensen/Poulsen/Reich [5], apparently unaware of Gillespie's

result [1] (or of any of the other papers below) proved a

stronger version in which "left endpoint" becomes "selection

via the function G(x,y)", where the domain of G is all pairs

of real numbers x < y in the integration interval [0,1] such

that x <= G(x,y) <= y and G satisfies the intermediate value

property in each variable (i.e. for each fixed p and q,

G(p,t) and G(t,q) satisfy the intermediate value property

relative to the variable t). Note that the choice of

G(x,y) = x gives the left endpoint formulation.

I believe Giovanni [2] (also apparently unaware of Gillespie [1])

proved a result similar to what Kristensen/Poulsen/Reich [5]

proved, but I think he restricted himself to the case where

G(x,y) is continuous (I don't know if this is continuity

in both variables or the weaker notion of continuity in

each variable). However, I haven't tried asking anyone

to translate anything from this paper, so I'm not

completely sure what he does. [Fund. Math. papers are

on the internet at <http://matwbn.icm.edu.pl> if you

want to take a look at it.]

Hildebrandt [3] (p. 273) wrote: "It seems probable that

[Gillespie's] proof can be carried over to the Stieltjes

integral [with respect to] a continuous g(x) of bounded

variation." I thought I had a paper somewhere, or at least

had seen such a paper at one time, that did this. However,

I couldn't find one, so this might make for a good undergraduate

or Master's thesis if you know of anyone (student or faculty

member) who needs a real analysis topic.

Zorn [8] (Lemma 1, p. 148) makes an interesting (to me,

at least) observation. The proof of Zorn's observation

is immediate and Zorn's paper is otherwise not particularly

related to the other papers I'm talking about, but I don't

think I've seen it pointed out anywhere else. Let's call

a bounded function on on a compact interval "weakly Cauchy

integrable" (my term) if, for each subinterval in a partition,

the evaluation point is either the left endpoint or the right

endpoint. Note that requiring a uniform choice of "left"

or "right" for all the subintervals puts us back to Cauchy's

formulation. To see that weakly Cauchy integrable implies

Riemann integrable, we simply observe that f(t_n)*(x_n - x_(n-1))

is equal to f(t_n)*(x_n - t_n) + f(t_n)*(t_n - x_(n-1)).

That is, use the "Riemann selection point t_n" to divide

the interval [x_(n-1), x_n] into two intervals, [x_(n-1), t_n]

and [t_n, x_n].

Kieffer/Stanojevic [4] study a certain weakening of the

selection of an arbitrary Riemann evaluation point in each

subinterval of a partition that, in a certain probabilistic

sense, results in almost sure convergence to (every?) Lebesgue

integrable functions. I haven't looked at it very much and

my background is rather weak in what is involved.

Tong [7] considers ways in which the convergence of equal

width subintervals is sufficient for Riemann integrability.

He was apparently unaware of Sklar [6], who seems to prove

very similar results.

[1] David Clinton Gillespie, "The Cauchy definition of a definite

integral", Annals of Mathematics (2) 17 (1915-16), 61-63.

[JFM 45.0441.02]

[2] Dantoni Giovanni, "Sul confronto di alcune definizioni di

integrale definito", Fundamenta Mathematicae 19 (1932), 29-37.

[Zbl 5.20002; JFM 58.0237.01]

[3] Theophil Henry Hildebrandt, "Definitions of Stieltjes integrals

of the Riemann type", American Mathematical Monthly 45 #5

(May 1938), 265-278. [Zbl 19.05604; JFM 64.0198.01]

[4] John C. Kieffer and Caslav V. Stanojevic, "The Lebesgue integral

as the almost sure limit of random Riemann sums", Proceedings of

the American Mathematical Society 85 (1982), 389-392.

[MR 83h:26015; Zbl 497.28007]

[5] Erik Kristensen, Ebbe Thue Poulsen, and Edgar Reich,

"A characterization of Riemann-integrability", American

Mathematical Monthly 69 #6 (June/July 1962), 498-505.

[MR 25 #4074; Zbl 113.04201]

[6] Abe Sklar, "On the definition of the Riemann integral", American

Mathematical Monthly 67 #9 (November 1960), 897-900.

[MR1530955]

[7] Jingcheng Tong, "Partitions of the interval in the definition

of Riemann's integral", International Journal of Mathematical

Education in Science and Technology 32 (2001), 788-793.

[MR1862677; Zbl 1048.26503]

[8] Max August Zorn, "Approximating sums", American Mathematical

Monthly 54 #3 (March 1947), 148-151.

[MR 8,450g; Zbl 29.03101]

Dave L. Renfro

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