Lebesgue versus Eudoxus : Metric Theory.

http://en.wikipedia.org/wiki/Mathematical_analysis

Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

http://en.wikipedia.org/wiki/Integral

Pre-calculus integration

The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus (ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. Similar methods were independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere (Shea 2007; Katz 2004, pp. 125–126).

The next significant advances in integral calculus did not begin to appear until the 16th century. At this time the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of xn up to degree n = 9 in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.
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Formalizing integrals

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann integrable on a bounded interval, subsequently more general functions were considered – particularly in the context of Fourier analysis – to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.

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Lebesgue integral
Main article: Lebesgue integration

Riemann–Darboux's integration (blue) and Lebesgue integration (red).

It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann integrable, and so such limit theorems do not hold with the Riemann integral. Therefore it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated (Rudin 1987).

Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel:
I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.
Source: (Siegmund-Schultze 2008)

As Folland (1984, p. 56) puts it, "To compute the Riemann integral of f, one partitions the domain [a,b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f". The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure μ(A) of an interval A = [a,b] is its width, b − a, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.

Using the "partitioning the range of f" philosophy, the integral of a non-negative function f : R → R should be the sum over t of the areas between a thin horizontal strip between y = t and y = t + dt. This area is just μ{ x : f(x) > t} dt. Let f∗(t) = μ{ x : f(x) > t}. The Lebesgue integral of f is then defined by (Lieb & Loss 2001)

where the integral on the right is an ordinary improper Riemann integral (note that f∗ is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral). For a suitable class of functions (the measurable functions) this defines the Lebesgue integral.

A general measurable function f is Lebesgue integrable if the area between the graph of f and the x-axis is finite:

In that case, the integral is, as in the Riemannian case, the difference between the area above the x-axis and the area below the x-axis:

where

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Other integrals

Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:
The Darboux integral which is equivalent to a Riemann integral, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. Darboux integrals have the advantage of being simpler to define than Riemann integrals.
The Riemann–Stieltjes integral, an extension of the Riemann integral.
The Lebesgue-Stieltjes integral, further developed by Johann Radon, which generalizes the Riemann–Stieltjes and Lebesgue integrals.
The Daniell integral, which subsumes the Lebesgue integral and Lebesgue-Stieltjes integral without the dependence on measures.
The Haar integral, used for integration on locally compact topological groups, introduced by Alfréd Haar in 1933.
The Henstock–Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.
The Itō integral and Stratonovich integral, which define integration with respect to semimartingales such as Brownian motion.
The Young integral, which is a kind of Riemann–Stieltjes integral with respect to certain functions of unbounded variation.
The rough path integral defined for functions equipped with some additional "rough path" structure, generalizing stochastic integration against both semimartingales and processes such as the fractional Brownian motion.
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Properties

I take Berkley's protestations with a pinch of salt! He clearly struggled to understand Newton and the Greek Philosophers from which he drew his inspiration. Berkely a cleric was more about attacking sciences drift away from clerical control than understanding why Newton relied on so called Pagan Philosophy!

You may see the games Mathematicians play to preserve a method. Eudoxus on the other hand was clear and simple: the arithmos or mosaic under a curve, or with a curved boundary was just that, a countable mosaic which could be constucted exhaustively. At some point we give up. At that point we have a set of dimensions for our units which we may call monads. We may now standardise the form and compute amd compare in a standard format.

Why?

Because some architect or engineer or artisan would like to be able to copy or make some image and they would like to know what proportions are involved, how much work they are going to have to do to achieve a certain standard of finish. This is the function of our processing. From it we can build and design synthetic forms and mechanical models efficiently.

The arithmoi have this role of mathematical clay.Pythagoras and Pythagoreans appreciated that. Without these drawings on the f;oor done with mosaic pieces you cannot have geometry, the measure of land!Astronomy, on the other hand used different forms based on triangles and spheres. These were solid forms imagined huge, and by imagination immaterial. These Arithmoi, the solid ones had their own calculus. Eudoxus clearly lays these out in Book 7, or rather Euclid does, but Book 6 is the necessary procedural precursors.

We are coerced to believe that a mathematical model will give deeper insight, but the reverse is truer. Deeper insight enables a mathematical model. Deeper insight enables the correct tool for measurement to be devised as ametron.monce I have a Metron I can plug it into a formal process called proportioning by Arithmoi. But before I get the correct Metron I can suppose, and these suppositions reveal my limitations. If I only have insight according to my limitations I will choose the wrong Metron, and I will reate a self fulfilling prophecy in the form of a mathematical model.

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