History of Lebesgue integration

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	Revision 13 . . (edit) November 22, 2001 7:57 pm by Gareth Owen
	Revision 12 . . November 22, 2001 1:06 pm by Chenyu *[started translating the article into English. It's not rigorous and may not be correct, but it's more comprehensible (but still needs work)]**
	Revision 11 . . November 22, 2001 12:41 pm by AxelBoldt
	Revision 10 . . November 22, 2001 12:17 pm by AstroNomer [not sure if this is right, but until a mathematician tells, i think it will do.]
	Revision 9 . . (edit) November 22, 2001 11:47 am by ManningBartlett [added talk]
	Revision 8 . . November 22, 2001 10:31 am by (logged).200.219.xxx [Some of what this entry said did not make any sense in the english language.]
	Revision 7 . . September 30, 2001 7:19 pm by (logged).29.241.xxx [Definition of Lebesgue integration]

Difference (from prior major revision) (minor diff, author diff)

Changed: 2c2,18

(Riemann-) integration? to more functions and to more general settings. The main advantage of the Lebesgue integral over the Riemann integral is that more functions become integrable, and that the integral can often be determined easily using convenient convergence theorems. Furthermore, the extension of the notion of integration to functions defined on general measure spaces, which include probability spaces, allows the proper formulation of the foundations of probability and statistics. A formal introduction of the concept follows.

(Riemann-)integration to more functions and to more general settings.

Integration
is the mathematical operation which corresponds to finding the area under
a function. A Riemann integral defines this operation by filling the area under the curve with smaller and smaller rectangles. As the rectangles become smaller and smaller, the total area of the rectangles becomes closer and closer to the area under the curve. Unfortunately, there are functions for
which this method of finding areas does not work, for example, consider a
function f(x) which is 0 when x is rational and 1 otherwise. You can't draw
rectangles under the curve and find its area using Riemann integral.

This is where Lebesque integration comes in. Instead of using limits, Lebesque integration uses maximums. Take that function f(x). I don't know
the area underneath that function, but I do know that it is greater than all of the functions which are smaller or equal to than the area of all of the functions which are smaller
than f(x) across the interval I am interested in. The idea behind Lebesque
intergration is to run through all of a set of functions which are smaller
or equal to f(x), and the upper bound of those functions is the area of the
function.

The main advantage of the Lebesgue integral over the Riemann integral is that more functions become integrable, and that the integral can often be determined easily using convenient convergence theorems. Furthermore, the extension of the notion of integration to functions defined on general measure spaces, which include probability spaces, allows the proper formulation of the foundations of probability and statistics. A formal introduction of the concept follows.