Lebesgue integral
Main article: Lebesgue integration 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 generalThe Riemann integral is not defined for a wide range of functions and situations of importance in applications (and of interest in theory). For example, the Riemann integral can easily integrate density to find the mass of a steel beam, but cannot accommodate a steel ball resting on it. This motivates other definitions, under which a broader assortment of functions is integrable (Rudin 1987). The Lebesgue integral, in particular, achieves great flexibility by directing attention to the weights in the weighted sum.
The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measurable; the volume or measure of the Lebesgue 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.
To exploit this flexibility, Lebesgue integrals reverse the approach to the weighted sum. 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".
One common approach first defines the integral of the indicator function In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A of a measurable set A by:
- .
This extends by linearity to a measurable simple function In mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line which attains only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are s, which attains only a finite number, n, of distinct non-negative values:
(where the image of Ai under the simple function s is the constant value ai). Thus if E is a measurable set one defines
Then for any non-negative measurable function f one defines
that is, the integral of f is set to be the supremum In mathematics, given a subset S of a partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to each element of S. Consequently, the supremum is also referred to as the least upper bound, lub or LUB. If the supremum exists, it may or may not belong to S. If the supremum exists, it is of all the integrals of simple functions that are less than or equal to f. A general measurable function f, is split into its positive and negative values by defining
Finally, f is Lebesgue integrable if
and then the integral is defined by
When the measure space on which the functions are defined is also a locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon measures In mathematics, a Radon measure, named after Johann Radon, on a Hausdorff topological space X is defined in measure theory to be a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular, of which the Lebesgue measure is an example) and integral with respect to them can be defined differently, starting from the integrals of continuous functions with compact support In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set. This concept is used very widely in mathematical analysis. In the form of functions with support that is bounded, it also plays a major part in various types of mathematical duality theories. More precisely, the compactly supported functions form a vector space A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields that carries a natural topology Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology, and a (Radon) measure can be defined as any continuous linear In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is in especially common use, for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides functional on this space; the value of a measure at a compactly supported function is then also by definition the integral of the function. One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function. This is the approach taken by Bourbaki (2004) and a certain number of other authors. For details see Radon measures In mathematics, a Radon measure, named after Johann Radon, on a Hausdorff topological space X is defined in measure theory to be a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular.
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