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'''Functional integration''' is a collection of results in [[mathematics]] and [[physics]] where the ___domain of an [[integral]] is no longer a region of space, but a [[Function space|space of functions]]. Functional integrals arise in [[probability]], in the study of [[partial differential equations]], and in the [[path integral formulation|path integral approach]] to the [[quantum mechanics]] of particles and fields.
In an ordinary integral (in the sense of [[Lebesgue integration]]) there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the ___domain of integration). The process of integration consists of adding up the values of the integrand for each point of the ___domain of integration. Making this procedure rigorous requires a limiting procedure, where the ___domain of integration is divided into smaller and smaller regions. For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the ___domain of integration is a space of functions. For each function, the integrand returns a value to add up. Making this procedure rigorous poses challenges that continue to be topics of current research. [Note on this introduction: While the main rigorous mathematical complications/issues with Functional/Feynman integral is directly linked to measure theory, instead of starting with Lebesgue integral, starting with Riemann integral (or even Newton /Leibnitz integral) would make this introduction far more accessible to wider audience before we go into topics like measure theory including Lebesgue integral, and functional space. Is anyone with a great communication skills willing to write up a comprehensive introduction that is accessible to 1st or 2nd year undergraduates in pure mathematics/mathematical physics or theoretical physics/physics?]
Functional integration was developed by [[Percy John Daniell]] in an article of 1919<ref>{{Cite journal
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