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== Introduction ==
The Lebesgue integral is a mathematical concept in the field of [[Real Analysis|real analysis]] and [[Measure Theory|measure theory]], named after French mathematician [[Henri Lebesgue|Henri Lebesgue]]. It extends the concept of the [[Riemann Integral|Riemann integral]] by providing a more general method of integration that can handle a broader class of functions and sets. The Lebesgue integral is a fundamental tool in the study of [[Probability Theory|probability theory]], [[Quantum Mechanics|quantum mechanics]], and other areas of mathematics and physics.

[[Image:Detail-61210.jpg|thumb|center|A mathematical equation representing the Lebesgue integral.|class=only_on_mobile]]
[[Image:Detail-61211.jpg|thumb|center|A mathematical equation representing the Lebesgue integral.|class=only_on_desktop]]

== Definition ==
The Lebesgue integral is defined in terms of [[Measure (mathematics)|measure]] and [[Measurable Function|measurable functions]]. Given a measurable function ''f'' defined on a set ''E'' with measure ''μ'', the Lebesgue integral of ''f'' with respect to ''μ'' over ''E'' is denoted as ∫''E f dμ''. The definition of the Lebesgue integral relies on the concept of [[Simple Function|simple functions]], which are functions that take on only a finite number of values.

== Properties ==
The Lebesgue integral has several important properties that distinguish it from other forms of integration. These include:

* '''Linearity''': The Lebesgue integral is linear, meaning that the integral of the sum of two functions is equal to the sum of their integrals, and the integral of a function multiplied by a constant is equal to the constant times the integral of the function.

* '''Monotonicity''': If one function is pointwise greater than or equal to another, then its integral is also greater than or equal to the integral of the other function.

* '''Positivity''': The integral of a nonnegative function is nonnegative.

* '''Dominant Convergence Theorem''': This theorem, also known as Lebesgue's Dominated Convergence Theorem, states that if a sequence of functions converges pointwise to a limit function, and if all the functions in the sequence are dominated by a single integrable function, then the integral of the limit function is equal to the limit of the integrals of the functions in the sequence.

== Applications ==
The Lebesgue integral has numerous applications in various fields of mathematics and physics. In probability theory, the Lebesgue integral is used to define the expected value of a random variable. In quantum mechanics, it is used in the formulation of the [[Schrödinger Equation|Schrödinger equation]]. In functional analysis, the Lebesgue integral is used in the definition of [[Lp Space|Lp spaces]], which are fundamental in the study of [[Partial Differential Equations|partial differential equations]] and [[Fourier Analysis|Fourier analysis]].

== Comparison with the Riemann Integral ==
The Lebesgue integral is more general than the Riemann integral and can handle functions and sets that the Riemann integral cannot. For instance, the Lebesgue integral can integrate functions that are discontinuous on sets of measure zero, while the Riemann integral cannot. Furthermore, the Lebesgue integral is more suited to limit processes, thanks to properties like the Dominated Convergence Theorem.

== Conclusion ==
The Lebesgue integral is a powerful tool in mathematics, extending the concept of integration to a broader class of functions and sets. Its properties and applications make it a fundamental concept in real analysis, measure theory, probability theory, quantum mechanics, and other areas of mathematics and physics.

== See Also ==
* [[Measure Theory]]
* [[Real Analysis]]
* [[Probability Theory]]
* [[Quantum Mechanics]]
* [[Functional Analysis]]

[[Category:Mathematics]]
[[Category:Real Analysis]]
[[Category:Measure Theory]]