Riemann Integral

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It is discontinuous on both the rationals and the irrationals. Every irrational is the limit of a Cauchy sequence of rationals. Every rational is the limit of a Cauchy sequence of irrationals. You do not need to know the measure of the rationals.
Are you getting at the measure of the irrationals is 1 over the interval [0,1]. Sure. Are you saying if we do not define, or define it, at the rationals, it doesn't matter which, the measure of the real line between 0 and 1 is the measure of the irrationals ie one. Again sure - just different language to what I said before. But how does that resolve the original query. Under the usual definition of the Riemann Integral it means if it is discontinuous at the rationals it is not Riemann integrable. Thus one could say it cant be discontinuous at the rationals because its Riemann Integrable. - so my argument since the rationals are dense it must be continuous at those points - hence f=g? Have I got your argument correct?

My suspicion, and I have never seen a development along those lines, Riemann Integration can be defined in a more general way as detailed previously.

Thanks
Bill
 
  • #27
lavinia
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Are you getting at the measure of the irrationals is 1 over the interval [0,1]. Sure. Are you saying if we do not define, or define it, at the rationals, it doesn't matter which, the measure of the real line between 0 and 1 is the measure of the irrationals ie one. Again sure - just different language to what I said before. But how does that resolve the original query. Under the usual definition of the Riemann Integral it means if it is discontinuous at the rationals it is not Riemann integrable. Thus one could say it cant be discontinuous at the rationals because its Riemann Integrable. - so my argument since the rationals are dense it must be continuous at those points - hence f=g? Have I got your argument correct?

My suspicion, and I have never seen a development along those lines, Riemann Integration can be defined in a more general way as detailed previously.

Thanks
Bill
Not sure how to answer your post.

The Dirichlet function is discontinuous on the entire interval not only the rationals. So it is not Riemann integrable. Every irrational is the limit of a Cauchy sequence of rationals and the limit of the Dirchlet function along this sequence is zero. But its value is 1 on every irrational so it is discontinuous at every irrational. This already means that it is not Riemann integrable. But it is also discontinuous on the rationals. Along any Cauchy sequence of irrationals that converges to a rational, the limit is 1, But the value of the Dirichlet function on each rational is zero. So the measure of the rationals doesn't come into the argument since the Dirichlet function is discontinuous everywhere.

The original post is answered with Mathwonk's hint which is that at points where g is continuous it must equal the function f. If g is Riemann integrable it is continuous except possibly on a set of measure zero. So it equals f almost everywhere. However g does not have to equal f exactly since for instance one can take g to be f everywhere except at a single irrational number.

Not sure if this is what you were asking.
 
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So the measure of the rationals doesn't come into the argument since the Dirichlet function is discontinuous everywhere.The original post is answered with Mathwonk's hint which is that at points where g is continuous it must equal the function f. If g is Riemann integrable it is continuous except possibly on a set of measure zero. So it equals f almost everywhere. However g does not have to equal f exactly since for instance one can take g to be f everywhere except at a single irrational number.
Got it.

Not sure if this is what you were asking.
Its cleared up now.

Thanks
Bill
 
  • #29
mathwonk
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@Svein: Yes the history of lebesgue versus riemann integration is a little confusing. Let me try to clarify my claim that riemann actually proved "lebesgue's criterion" even before lebesgue stated it. Lebesgue indeed defined the concept of "measure zero" to mean a set that can be covered by a countable sequence of intervals whose total length is as small as desired, i.e. a set has measure zero if: given any epsilon e > 0, there is a countable union of intervals that cover the set, the sum of whose lengths is less than e.

there is however a related concept now called "content zero" which says that for every epsilon, there is a finite sequence of intervals that cover the set, and whose lengths total less than epsilon. Riemann proved that a function f is Riemann integrable if and only if for every epsilon, and every delta, there is a finite sequence of intervals, of total length less than epsilon, that cover the set where f has oscillation greater than delta. I.e. he proved, in modern language, that f is integrable iff for every delta, the set where f has oscillation greater than delta, has content zero. Since it is elementary that the set where f is discontinuous is the union over n of the sets where f has oscillation > 1/n, and since a countable union of sets of content zero trivially has measure zero, Riemann's criterion, although not stated using the word "measure", nonetheless trivially implies Lebesgue's criterion. Thus Lebesgue merely restated Riemann's criterion in a different wording, but offered no new insight whatsoever as to when a function is Riemann integrable.

This is apparently not widely understood, especially in the US, since Riemann's works were only recently translated into English, but I was fortunate enough to be the official reviewer of that translation for Math Reviews, hence was one of the first to read it. I have also seen Riemann's result misstated in some textbooks by apparent "experts". If you have any further interest, I invite you to read the English translation, or the original if you read German, for yourself. It is slow going, but very rewarding reading.

Indeed I have just read the wikipedia article you linked, and while in general wikipedia articles are not always reliable, this one does cite the original article of Riemann as follows:

......(On the concept of a definite integral and the extent of its validity), pages 101–103.

Notice the phrase "and the extent of its validity", (und den Umfang seiner Gultigkeit, for fresh, with apologies). That part is indeed the criterion I have stated as to exactly when the Riemann integral works. I have just now consulted the article itself again and noted for the first time, that Riemann also gives a criterion for an unbounded function to be integrated by taking limits.

Wow! I have very few calculus books left on my shelf but one that I have saved is the excellent one by the magnificent author Joseph Kitchen, Calculus of One Variable. I have consulted it just now on this topic and found for the first time ever, an author who is thoroughly familiar with exactly what I have just explained. On pages 357-362 Kitchen explains in detail Riemann's criterion for integrability, in exactly the language (of content) that I have used, and then proves (in about 8 lines, i.e. trivially) that it is equivalent to Lebesgue's criterion, (by exactly the same argument I myself gave when reading Riemann). Kitchen's book was published over 50 years ago(!), but apparently did not become a standard, as much as the other famous honors books from the 1960's by Spivak and Apostol.

As a personal note, Kitchen was an extremely popular instructor at Harvard when I was a freshman there in 1960-61, and I recall the reviews of his course were summarized roughly as follows: "A large minority of Professor Kitchen's students are convinced that he is God".

Remark: Although his discussion persuades me of his thorough familiarity with the work of Riemann, he gives no specific historical reference to either Riemann's or to Lebesgue's works for this criterion, except for using their names on the concepts.
 
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