What is Measure theory: Definition and 119 Discussions

In mathematics, a measure on a set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the usual length, area, or volume to subsets of a Euclidean spaces, for which this be defined. For instance, the Lebesgue measure of an interval of real numbers is its usual length.
Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see § Definition, below). A measure must further be countably additive: if a 'large' subset can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets that are measurable, then the 'large' subset is measurable, and its measure is the sum (possibly infinite) of the measures of the "smaller" subsets.
In general, if one wants to associate a consistent size to all subsets of a given set, while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

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  1. P

    I On transformation of r.v.s. and sigma-finite measures

    I'm reading this article on transformation of random variables, i.e. functions of random variables. We have a probability space ##(\Omega, \mathcal F, P)## and measurable spaces ##(S, \mathcal S)## and ##(T, \mathcal T)##. We have a r.v. ##X:\Omega\to S## and a measurable map ##r:S\to T##. Then...
  2. P

    I On pdf of a sum of two r.v.s and differentiating under the integral

    I'm reading in my book about the pdf of the sum of two continuous random variables ##X_1,X_2##. First, I'm a bit confused about the fact that the sum of two continuous random variables may not be continuous. Does this fact make the derivation below still valid or is there some key assumption...
  3. P

    I On theorem 1.19 in Folland's and completion of measure

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  4. cianfa72

    I ##L^2## square integrable function Hilbert space

    Hi, I'm aware of the ##L^2## space of square integrable functions is an Hilbert space. I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral...
  5. P

    I On integral of simple function and representation

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  6. P

    I Collection of finite unions of half-open intervals form an algebra

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  7. P

    I Parallel rectangles contained in oblique rectangle

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  8. P

    I Help with basic transfinite induction proof

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  9. P

    I Two basic results from measure theory -- on volumes of rectangles

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  10. Lagrange fanboy

    I Prove projection of a measurable set from product space is measurable

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  11. L

    Approaching the Measure of a Set: Strategies for Finding f(Eα)

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  12. U

    I Limit of limits of linear combinations of indicator functions

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  13. A

    Analysis Prerequisites Measure theory for ug student in physics

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  14. MathematicalPhysicist

    I A claim in measure theory which seems flawed to me

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  15. D

    A Applications of analysis in signal processing/machine learning?

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  16. Jatex

    I Radon-Nikodym Derivative and Bayes' Theorem

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  17. C

    Courses Graduate level Mathematics courses of interest for Biological Physics

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  18. L

    A Definitions of Cylinder Sets and Cylinder Set Measure

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  19. H

    Is Tonelli's Theorem a Useful Tool for Determining the Existence of Integrals?

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  20. J

    MHB Measure Theory - Existence of Fsigma set contained in measurable set

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  21. Bill2500

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  22. Bill2500

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  23. L

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  24. Joppy

    MHB Reference request - Measure theory

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  25. pawlo392

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  26. G

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  27. K

    Is (-infinity, b) an event for any real number b?

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  28. Luck0

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  29. D

    Show a limited function is measurable

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  30. G

    Geometry What would a textbook on measure theory be called?

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  31. Demystifier

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  32. mr.tea

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  33. DavideGenoa

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  34. B

    I Extensive properties as measures

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  35. DavideGenoa

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  36. Jianphys17

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  37. M

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  38. T

    Proving Compact Set Exists with m(E)=c

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  39. T

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  40. L

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  41. A

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  42. J

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  43. M

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  44. M

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  45. A

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  46. A

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  47. A

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  48. A

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  49. M

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  50. Barioth

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