In mathematics, Lebesgue integration refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a sub-domain of the real line or a higher dimensional Euclidean space with respect to the Lebesgue measure. This article focuses on the more general concept.
Lebesgue integration plays an important role in real analysis, the axiomatic theory of probability, and many other fields in the mathematical sciences.
The integral of a non-negative function can be regarded in the simplest case as the area between the graph of that function and the x-axis. The Lebesgue integral is a construction that extends the integral to a larger class of functions defined over spaces more general than the real line.
For non-negative functions with a smooth enough graph (such as continuous functions on closed bounded intervals), the area under the curve is defined as the integral and computed using techniques of approximation of the region by polygons (see Simpson's rule). For more irregular functions (such as the limiting processes of mathematical analysis and probability theory), better approximation techniques are required in order to define a suitable integral.
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