As a researcher, often, it comes to my mind why do we require measure theory and what is the significance and applications of measure theory, measurable set, space and function in real life. Can anyone help me and enlighten me in this direction.
Measure theory is a branch of mathematics that provides a foundation for the study of integration and probability theory. It is used to define and analyze notions of size, length, and volume, among other concepts, in abstract mathematical spaces.
The actual meaning and significance of measure in mathematics can be understood as a function that assigns a size or magnitude to sets of points in a mathematical space. This size can be thought of as a generalization of the notion of length, area, or volume in familiar Euclidean spaces, and can be used to define concepts such as integrals, probability distributions, and stochastic processes.
Measure theory is a crucial tool in many areas of mathematics and its applications, including real analysis, functional analysis, probability theory, and geometric measure theory. It provides a rigorous framework for defining and working with mathematical objects and concepts that are difficult to handle with more traditional techniques.
In summary, measure theory is important in mathematics because it provides a framework for defining and analyzing abstract concepts of size and magnitude, and it is essential for the development and application of many mathematical theories and techniques, including those in real analysis, probability theory, and geometric measure theory.
In probability theory, we often work with sample spaces, which are sets of outcomes of a random experiment. For example, the sample space of a coin flip might be {heads, tails}. In order to assign probabilities to events in the sample space, we need a way to measure the size of the events. In other words, we need to assign a measure to the events.
@Rodolfo Bolaños Barrera, could you please send some material related to it to me
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"Real Analysis" by H.L. Royden
"Measure, Integration and Function Spaces" by N.L. Carothers
"A Course in Measure Theory" by Terence Tao
"Introduction to Measure Theory" by Terence Tao
"Lebesgue Integration on Euclidean Space" by Frank Jones
"Measure, Integration and Probability" by Richard L. Wheeden and Antoni Zygmund
"Measures, Integrals and Martingales" by René L. Schilling, Renato S. Cruz, and Lijing Sun
"Lebesgue Measure and Integration: An Introduction" by Frank Burk
"Introduction to Real Analysis" by Robert G. Bartle
"Measures, Integrals and Applications" by George G. Robertson, Samuel Kuo, and Arnold W. Miller
Thanks a lot everyone. I started to chase my query and found some answers. Measure theory deals with measures, now the need of measure is to extend in general the concept of length function to all subsets of R, which primarily is subjected to the open and closed sets. Afterwards, the notion of measure, outer measure and lebesgue measure came into existence and the theory of integrals are extended to lebesgue theory of integrals.
Measurement theory is a vital part of scientific methodology. In this sense, it is of key importance that we do understand what we really measure, when we measure (facts vs. artifacts). There exists a great lack of understanding the exact foundations of science and its practical consequences; such methodical questions are especially crucial for the scientific progress of medical diagnostics, psychological assessments and real-world economics, due to the nature of the subject (living vs. non-living matter).
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