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MathWorld Book. Wolfram Web Resources ». Created, developed, and nurtured by Eric Weisstein at Wolfram Research. Wolfram Alpha » Explore anything with the first computational knowledge engine. As the title suggests, the course uses probability as the natural motivation for measure theory, which I think is a pretty good idea. As I now understand is quite different to how things are in the US, I've never taken a 'graduate real analysis'-type class.
To comment on the discussion following fedja's answer, the book for the course is really the first half of David Williams' Probability with Martingales and is a good English-Language intro to measure theory and isn't too long. Unfortunately, it's called Probability with Martingales so all this isn't terribly obvious, though I've needed no other basic measure theory text and am not a probabilist myself. The pedagogical aim would be to convince students that measure theory is a good setting for a rigorous theory of probability, which, before that stage, will only have been taught naively.
There are plenty of explainable things that could help here: Why can't all sets be measurable? Why do we need countable additivity? What's the deal with "almost surely"? The need for a rigorous theory of expectation, conditional expectation and various notions of convergence of random variables and so forth leads to the Lebesgue integral relatively naturally, if you ask me.
Integrals of simple functions are basically just expectations of Bernoulli random variables If monotonicity of measures is understood, then the monotone ,and hence dominated, convergence theorems some of Lebesgue's main bonuses over Riemann start seeming more natural. I think if you want to make sense of and motivate numerically integrating a function which is done almost always in numerical analysis, I believe Riemann integration is a better way to motivate than Lebesgue.
Also, Riemann integration helps to provide a motivation for Lebesgue integration as to why we need to define Lebesgue integration in the first place.
In introductory calculus classes, I work hard to get my students to understand the heuristic: break complicated problem into simple close enough to constant pieces; find the answer to the simple pieces; add the results using an integral. This is how they can understand the presence of integrals in many contexts. Formally, of course, this is a shorthand for Riemann integration. Although it's true that the Lebesgue integral is technically more complicated than the Riemann integral, I don't think that's really the point.
In fact, the Riemann integral is pretty technical already, as far as most undergraduates are concerned. The real point is the "fundamental theorem of the calculus", which is at least conceptual and amenable to a really good plausibility argument in the case of the Riemann integral, but requires some genuinely deep analysis in the case of the Lebesgue integral.
Don't get me wrong -- I really love integration theory. And I've never taught serious undergraduate analysis, so I don't have any actual experience in this area.
I do have a funny story, however:. I first learned "real analysis" from a course taught by Richard Brauer. Brauer was of course an algebraist, but at the time, he taught in consecutive years the basic graduate "real analysis", "complex analysis" and algebra courses. He was as close as I felt I would ever get to the great tradition of mathematicians who had a broad view of the field as a whole.
Of course the book we used as a reference for measure theory and integration was Halmos. At one point, he remarked, "Halmos is a curious book. When you finish this book you will have a very good understanding of how to integrate the function 1. Of course he was referring to the fact that Halmos doesn't even touch on the question of differentiation in Euclidean space, and so doesn't really deal with the "fundamental theorem".
I found that very elegant, my students found that difficult to digest. They were familiar with the Riemann integral but not so much with abstraction in mathematics. And then do more abstract measure theory later, never mentioning the Riemann integral?
I have no idea. I think using this definition is easy and geometrically intuitive, and on the other hand working with this definition prepares you conceptually for the Lebesgue integral where you juggle "simple" functions instead of step functions. Thus, you already get a nice piece of the Lebesgue point of view.
Similarly for the other inequality. Thus, one can go pretty far with this definition, but the results which separate the Riemann integral from the Lebesgue integral e. You could think this feature is either a clarification or a disadvantage. One certain disadvantage is that it is not the right point of view for integrating vector-valued functions. So you might decide it's better to define the integral in terms of Riemann sums in the first place giving more of a "metric space" point of view and less of an "ordered space" point of view.
Or you might even decide to skip some of these other topics, depending on your point of view and time available. The Riemann integral has a good geometrically motivated definition. So one teaches it but disadvantages are that even in elementary analysis it is not a complete tool. Lebesgue integral has the advantage that it is defined in a general set up and can handle multiple integration very well.
It is reasonably intuitive but not as intuitive as Riemann. But the gauge integral Henstock-Kurzweil covered nicely in C. Swartz book is intuitive and a nice generalization of Riemann. Every derivative is integrable. There are no improper integrals. All improperly integrable functions are gauge integrable as well.
We just use a gauge rather than mesh of a partition or a NET based o refinements. The integral is super Lebesgue yet definition is almost verbatim similar to Riemann integrable We can have a nice uniform convergence theorem. For graduate courses one can have The usual dominated convergence theorem slightly more general version than for Lebesgue integral. Further the differential calculus in Banach spaces can be carried out without any restrictive assumption like continuity of derivative and we can have nice integral version of mean value theorem.
On real line the integral is the perfect integral a satisfactory theory exists on euclidean spaces as well. Professor Buck in his book "Garden of integrals Terms this integral as integral of 21 st century. In many first-year calculus courses, one should not teach Riemann integration nor Lebesgue integration nor any other theory of integration. An integral is an area under a curve, but also any of various other sums of infinitely many infinitely small quantities. That's half of the Fundamental Theorem, and the other half can be given later, after one begins to talk about integrals.
The product rule is a corollary: think of a rectangle with two moving sides. So are some other things that are less important but are good exercises. Of course, differential calculus is not integration theory.
But I would not over-teach and use tools where they are far beyond the needs. So the choice depends, as usual, on our goal :. I guess one point that hasn't been made is that Riemann integral may have pedagogical value precisely because it's awkward and difficult. In my experience, this is precisely what makes it the most difficult definition in the analysis course, and even reasonable students struggle with it.
Usually, even the treatment of Lebesgue integral does not contain conditions that are logically that complicated - rather, the definition is split into many simple steps. But, at some point, better earlier than later, math students have to learn how to make sense of complicated statements with many nested quantifiers. Why not when learning Riemann integral? When I was introduced to measure theory, the professor chose to use the Choquet integral to obtain the Lebesgue integral.
An this uses the "good old" Riemann integral to integrate the pseudo-inverse of the cumulative distribution function I think, it was this book. As a student I enjoyed this approach because I really knew what the Riemann integral was about and also I had an understanding of the problems with it - but I was really confused by the way we had had the Lebesgue integral at the first place.
Riemann integration is still the simplest form of integration to introduce at an elementary calculus level. Moreover, while Lebesgue integration is often called a "generalization" of it, in important ways it could be considered as a sort of generalized "cousin" than its most direct generalization.
You see - in elementary calculus, there are at least two conceptually distinct ways of approaching the integral: the "antidifferentiation" approach, and the "area" approach. Both of these can be used to motivate the Riemann integral, but when you dig into them more deeply, they actually can be seen to diverge. This fits in with a general theme that one sees as one explores mathematics further - things that "seem" equivalent at first actually depend on certain assumptions for that equivalence, and one can talk about what happens when they break down.
For example, when one goes from classical to intuitionistic logic, one loses the equivalence of certain formulations of properties of the real numbers such as their completeness. Riemann integration most directly , I'd say, formulates the "antidifferentiation" approach. Here is how. Forget about tangent lines and all that mumbo-jumbo which is badly introduced anyways, and throw out the "standard textbook" presentation - its confusing, unintuitive bs.
Intuitively, the derivative represents the sensitivity of the output value of a function to a small perturbation applied to the input value, when the input value is set at some particular position. Think about, say, a knob on a mixing board, controlling how it changes the sound coming in. The knob can be considered as an input, and the sound produced as the output. Now consider what happens if you wiggle the knob a little bit back and forth around that set position, but don't actually move it to a greatly different location permanently.
How dramatically does the sound change? Moreover, how dramatically does it change compared to how much you wiggle it?
The derivative of a function is just the function that gives the sensitivity rating at every possible value of its input: that is, it tells me how much the "sound" changes for each and every fixed setting of the knob about which we wiggle it.
So that is how Riemann integration arises from "antidifferentiation". As you can see, the notion of "area" does not enter - instead, what we're actually trying to do is to reconstruct a function from its derivative. What should really be "surprising", in a sense, is that we can then also likewise develop the same formula for the integral starting from the concept of area, by the familiar "divide into vertical rectangles" construction, and the two happen to be equal, or equivalent. We could, perhaps, instead of calling this "integral", we could call it "area finder", and we have the intriguing result that "antidifferentiation" and the "area finder" tell us the same thing.
We then decide to name these as two different concepts of integration. And when we go on to more advanced settings, we find they generalize in different ways. The Lebesgue integral, naturally, "doesn't care" because instead of taking an upper and lower bound for which the direction from one to the other matters, it takes an undirected set as its "bounds".
This is what we'd expect from an area measure, and that is what it provides. Instead, the "correct" generalization of the Riemann integral in its own spirit is actually the theory of differential forms and instead of moving to "strange and pathological discontinuous functions", we move to differentiable manifolds and the integration of a function along a line on such a manifold.
We can later remarry the two theories, to then talk about pathological functions on a manifold, but they have to go their separate ways first. Hence, why that Riemann integration should be taught. You couldn't make sense out of any of these bodies of maths.
Likewise for other integration concepts. They all have uses and it is important to be able to both understand the relationships as well as the distinctions between them. The converse is false. There is no obvious reason we should go to the trouble of replacing all the integration results in our textbooks with weaker results that are no easier to prove. If the students are supposed to do only pure mathematics later on in their professional life, then I have nothing to say.
But if they are ever supposed to do some applied mathematics, physics or statistics, then my humble opinion is that one should never teach them only the Lebesgue integral. For instance, the connection between Riemann sums and numerical integration techniques rectangles, trapezes, Simpson, Romberg, Gaussian quadrature, sparse grids, etc.
But I'd like to mention another more fundamental and original reason:. Measure theory itself disqualifies the Lebesgue integral on which it relies at least for some important problems of applied probability theory.
Very shortly. Consider for instance the Behrens-Fisher problem, the most famous 90 years old open problem in statistics and applied probability theory, with vital applications such as clinical trials e. Lebesgue negligible null hypothesis testing problems. Therefore, the Behrens-Fisher problem with continuous parameters admits a trivial but totally useless and meaningless solution under measure theory on uncountable sets. That's the reason why the standard Bayesian solution to this problem, found in any textbook, is plain wrong.
In particular, it violates measure theory on uncountable sets, by assigning non-zero probabilities to Lebesgue-negligible e. See A fully Bayesian solution to k-sample tests for comparison and the Behrens-Fisher problem based on the Henstock-Kurzweil integral.
To get the correct, meaningful, useful and practical solution to this problem, we have to forget measure theory on uncountable sets, that is we have to forget actual, uncountable infinity and go back to potential infinity and measure theory on finite sets, in order to take the limit of the solutions to the discrete problems. Conversely, I consider that the Behrens-Fisher problem remained unsolved for such a long time until there is evidence to the contrary due to the old habit of jumping directly into the actual, uncountable infinity within the standard Borel-Lebesgue-Kolmogorov measure-theoretic setting for probability theory.
The Riemann integral is defined in terms of Riemann sums. Consider this image from the Wikipedia page :. We approximate the area under the function as a sum of rectangles. We can see that in this case, the approximation gets better and better as the width of the rectangles gets smaller.
In fact, the sum of the areas of the rectangles converges to a number, this number is defined to be the Riemann integral of the function. Note however that we can draw these rectangles in a number of ways, as shown below from this webpage. Maybe an historical perspective will help. It never hurts to have many points of view on what a mathematical definition is doing. For most of the 18th century the integral was considered just an antiderivative, much the way that many calculus students still consider it.
For continuous functions this is easy to arrange. The end result is that Cauchy proved that the integral of every continuous function could be approximated by these Riemann sums. Are there other functions not just continuous that also have an integral using Cauchy's same method. So the class of Riemann integrable functions is the class of functions for which Cauchy's method works.
This is somewhat larger than the class of continuous functions, large enough a class that nineteenth century mathematicians thought that they had a pretty good theory of integration. They didn't.
You'll have to look up what an upper sum and lower sum are. An upper sum intuitively is an approximation of the area of the curve from above, while a lower sum is an approximation of the area of the curve from below. In other words, the best approximation of the area of the function from below equals the best approximation of the area of the function from above. To answer this question, I will assume you already know what the integral of a step function is.
First I will answer with notation, and then give an equivalent formulation without notation. Without notation: A function on a closed bounded interval is called Riemann-integrable if there are two step functions respectively above and below it, and these two step functions can always be chosen so as to make the area between their graphs less than any specified quantity. If this limit exists, then the function is said to be Riemann integrable and the value of the Riemann integral is the limit the sums approach.
This representative set is necessary because riemann integrality is going to ultimately depend intuitively on a bunch of little rectangles. In computation, we pick either the left side of each rectangle or the right side in a Riemann sum. But, actually to prove integrability, we need to go further and say that the function's integral converges to a value, independently of how we choose these rectangles. Further, the mesh is important, because the rectangles need not be of equal width , all we need is that all of them are sufficiently "skinny.
This is the maximum amount of generality that I know of to show that something is Riemann integrable without invoking upper and lower sums. It does not depend on our choice of "rectangles", since the representative set is arbitrary, and it only requires that the "widest" rectangle is smaller than some real number.
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