Definition of integral

睇見上面兩個答案, 睇唔過眼......... @@

The R-S Integral and Lebesgue integral are defined in a totally different way.

Lebesgue integral is more general than R-S Integral, which is a general case of the usual Riemann Integral.

For R-S Integral Int fdg(x), the definition is given by the limit of the SUM f(t)[g(xi) - g(x_i-1)], (x_i-1 < t < x_i) ----- just like the usual Riemann integral, where g(x) = x
If the limit exists, (or more generally, the sup and inf over all partitions of the sum are the same) then f is integrable.

For Lebesgue Integral, we have a different way to deal with integral. We use approximation of simple functions. Basically, we have a measure on the space, and every positive measurable function can be approximated by an increasing sequence f_n = Sum aiX_Ai of finite linear combination of characteristic function of some measurable set Ai. (these are called simple functions).

The integral of simple functions are defined in an obvious way (Int X_A = m(A) where A is the measurable set), and for general function we use Monotone Convergence Theorem to define the integral as a limit, if the limit exists. These functions are then called Integrable. Finally, for arbitrary function, we split the function as the positive part and negative part f = f+ + f-, and evaluate the integral separately.

For quick reference you can look at PlanetMath:
http://planetmath.org/encyclopedia/RiemannStieltjesIntegral.html
http://planetmath.org/encyclopedia/Integral2.html

Also, most real analysis book will deal with these definitions, like Rudin, Royden and Folland.

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Improper integral only appear for Riemann Integral. It does not appear in Lebesgue Integral. Basically, it depends on the function in question. We calculate the integral of the function by approximation over compact intervals which are bounded away from the singular points. That's why we have those limit equation. Although we may get a value using this method, we may have different result (or divergence) when integrating over the same intervals with Lebesgue Integral (in this case the function is not integrable in the Lebesgue sense)

In calculus, the integral of a function is an extension of the concept of a sum. The process of finding integrals is called integration. The process is usually used to find a measure of totality such as area, volume, mass, displacement, etc., when its distribution or rate of change with respect to some other quantity (position, time, etc.) is specified. There are several distinct definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.
The term "integral" may also refer to antiderivatives. Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.
The integral of a real-valued function f of one real variable x on the interval [a, b] is equal to the signed area bounded by the lines x = a, x = b, the x-axis, and the curve defined by the graph of f. This is formalized by the simplest definition of the integral, the Riemann definition, which provides a method for calculating this area using the concept of limit by dividing the area into successively thinner rectangular strips and taking the sum of their areas (for example see this applet).
Alternatively, if we let


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then the integral of f between a and b is a measure of S. In intuitive terms, integration associates a number with S that gives an idea about the 'size' of the set (but this is distinct from its Cardinality or order). This leads to the second, more powerful definition of the integral, the Lebesgue integral.
Leibniz introduced the standard long s notation for the integral. The integral of the previous paragraph would be written
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sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating known as the integrand, and dx is a notation for the variable of integration. Historically, dx represented an infinitesimal quantity, and the long s stood for "sum". However, modern theories of integration are built from different foundations, and the notation should no longer be thought of as a sum except in the most informal sense. Now, the dx represents a differential form.
As an example, if f is the constant function f(x) = 3, then the integral of f between 0 and 10 is the area of the rectangle bounded by the lines x = 0, x = 10, y = 0, and y = 3. The area is the width of the rectangle times its height, so the value of the integral is 30. The same result can be found by integrating the function, though this is usually done for more complicated or smooth curves.




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Finding the area between two curves.ed or smooth curves.