MA359-15 Measure Theory Notes | Assignment Help | Syllabus
Country: United Kingdom
University: The University Of Warwick
The contemporary concept of measure, which emerged in the late 1800s, is a development of the concepts of length, area, and volume. A measure m is a natural extension of the following concepts: A measure m is a rule that gives a number to specific subsets A of a given space. 1) the length of an interval, 2) the area of a planar figure, 3) the volume of a solid, 4) the quantity of mass contained in a location, 5) the likelihood of an event from A occurring, and so on.
It has its origins in real analysis and is currently utilized in a variety of fields of mathematics, including geometry, probability theory, dynamical systems, functional analysis, and so on.
The integral of appropriate real valued functions with respect to m may be defined given a measure m. The Riemann integral is used to calculate continuous functions or functions with minimal discontinuities. The Riemann integral does not make sense for measurable functions that may be discontinuous ‘nearly everywhere.’ However, one of the fundamental concepts of contemporary analysis, Lebesgue’s integral (integral with regard to Lebesgue’s measure), may be defined in a more flexible and powerful way.
Aims of the module
Introduce the concepts of measure and integral with regard to a measure, demonstrate their fundamental characteristics, and provide the groundwork for future research in Analysis, Probability, and Dynamical Systems.
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