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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 willing to write up a comprehensive introduction that is accessible to 1st or 2nd year undergraduates?]
Functional integration was developed by [[Percy John Daniell]] in an article of 1919<ref>{{Cite journal
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