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| − | "Gaussian random variables and processes always played a central role in the probability theory and statistics. The modern theory of Gaussian measures combines methods from probability theory, analysis, geometry and topology and is closely connected with diverse applications in functional analysis, statistical physics, quantum field theory, financial mathematics and other areas." {{Cite|L}}
| + | ==Relations to Lebesgue integral== |
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| − | "The modern theory of Gaussian measures lies at the intersection of the theory of random processes, functional analysis, and mathematical physics and is closely connected with diverse applications in quantum field theory, statistical physics, financial mathematics, and other areas of sciences. The study of Gaussian measures combines ideas and methods from probability theory, nonlinear analysis, geometry, linear operators, and topological vector spaces in a beautiful and nontrivial way." {{Cite|B|Preface}}
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| − | "Gaussian processes have a rich, detailed and very well understood general theory, which makes them beloved by theoreticians.
| + | It is possible to treat absolutely convergent series as a special case of Lebesgue integrals. To this end, the countable set $\{0,1,2,\dots\}$ is treated as a [[measure space]]; its σ-algebra consists of all subsets, and its measure is the ''counting measure'' |
| − | In applications [...] it is important to have specific, explicit formulae that allow one to predict, to compare theory with experiment, etc. As we shall see [...] it will be only for Gaussian (and related [...]) fields that it is possible to derive such formulae in the setting of excursion sets." {{Cite|AT}}
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| − | {|
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| − | |valign="top"|{{Ref|B}}|| V.I. Bogachev, "Gaussian measures", AMS (1998). {{MR|}} {{ZBL|0913.60035}}
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| − | |valign="top"|{{Ref|L}}|| R. Latała, "On some inequalities for Gaussian measures", Proceedings of the International Congress of Mathematicians (2002), 813-822. {{MR|}} {{ZBL|1015.60011}}
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| − | |valign="top"|{{Ref|AT}}|| R.J. Adler, J.E. Taylor, "Random fields and geometry", Springer (2007). {{MR|}} {{ZBL|1149.60003}}
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Revision as of 17:40, 21 July 2012
Relations to Lebesgue integral
It is possible to treat absolutely convergent series as a special case of Lebesgue integrals. To this end, the countable set $\{0,1,2,\dots\}$ is treated as a measure space; its σ-algebra consists of all subsets, and its measure is the counting measure
How to Cite This Entry:
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=27152