Math 2.0

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Measure and integration

361 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 204 of its 206 ideas have a page so far.

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Map of the subject

Set systems 14 of 14

  1. Sigma-algebra definition
  2. Measurable space definition
  3. Algebra of sets definition
  4. Ring of sets definition
  5. Semiring of sets definition
  6. Pi-system definition
  7. Dynkin system definition
  8. Dynkin's π-λ theorem statement
  9. Monotone class theorem statement
  10. Borel sigma-algebra definition
  11. Borel hierarchy concept
  12. Analytic set definition
  13. Universally measurable set definition
  14. Standard Borel space definition

Measures 21 of 21

  1. Measure definition
  2. Measure space definition
  3. Countable additivity definition
  4. Finitely additive measure definition
  5. Continuity of measure from below and above statement
  6. Finite measure definition
  7. Sigma-finite measure definition
  8. S-finite measure definition
  9. Probability measure definition
  10. Dirac measure example
  11. Counting measure example
  12. Null set definition
  13. Almost everywhere definition
  14. Complete measure definition
  15. Completion of a measure definition
  16. Atom definition
  17. Non-atomic measure definition
  18. Sierpiński's theorem on non-atomic measures statement
  19. Pushforward measure definition
  20. Restriction of a measure definition
  21. Measure algebra definition

Construction of measures 19 of 20

  1. Outer measure definition
  2. Carathéodory's criterion statement
  3. Carathéodory's extension theorem statement
  4. Hahn–Kolmogorov theorem statement
  5. Premeasure definition
  6. Content definition
  7. Lebesgue measure object
  8. Lebesgue–Stieltjes measure definition
  9. Jordan measure definition
  10. Inner measure definition
  11. Metric outer measure definition
  12. Vitali set counterexample
  13. Non-measurable set concept
  14. Banach–Tarski paradox counterexample
  15. Hausdorff paradox counterexample
  16. Smith–Volterra–Cantor set example
  17. Steinhaus theorem statement
  18. Banach measure problem problem
  19. Measure problem of Lebesgue problem
  20. Littlewood's three principles concept

Measurable functions 13 of 13

  1. Measurable function definition
  2. Simple function definition
  3. Indicator function definition
  4. Egorov's theorem statement
  5. Lusin's theorem statement
  6. Convergence in measure definition
  7. Riesz's subsequence theorem statement
  8. Strongly measurable function definition
  9. Pettis measurability theorem statement
  10. Essential supremum definition
  11. Distribution function of a measurable function definition
  12. Layer cake representation statement
  13. Decreasing rearrangement definition

The Lebesgue integral 20 of 20

  1. Lebesgue integral definition
  2. Integral of a non-negative function definition
  3. Integrable function definition
  4. Monotone convergence theorem statement
  5. Fatou's lemma statement
  6. Dominated convergence theorem statement
  7. Uniform integrability definition
  8. Vitali convergence theorem statement
  9. Scheffé's lemma statement
  10. Riemann–Lebesgue lemma statement
  11. Absolute continuity of the Lebesgue integral statement
  12. Bochner integral definition
  13. Pettis integral definition
  14. Daniell integral definition
  15. Lebesgue–Stieltjes integration definition
  16. Jensen's inequality for integrals statement
  17. Hölder's inequality statement
  18. Minkowski inequality statement
  19. Chebyshev's inequality statement
  20. Markov's inequality statement

Lp spaces 13 of 13

  1. Lp space definition
  2. Riesz–Fischer theorem statement
  3. Density of simple functions in Lp statement
  4. L-infinity space definition
  5. Weak Lp space definition
  6. Lorentz space definition
  7. Orlicz space definition
  8. Riesz–Thorin theorem statement
  9. Marcinkiewicz interpolation theorem statement
  10. Dual of Lp statement
  11. Dunford–Pettis theorem statement
  12. Young's convolution inequality statement
  13. Convolution definition

Product measures 10 of 10

  1. Product measure definition
  2. Fubini's theorem statement
  3. Tonelli's theorem statement
  4. Product sigma-algebra definition
  5. Infinite product measure definition
  6. Kolmogorov extension theorem statement
  7. Ionescu-Tulcea theorem statement
  8. Cavalieri's principle statement
  9. Markov kernel definition
  10. Disintegration theorem statement

Signed measures and differentiation of measures 17 of 17

  1. Signed measure definition
  2. Complex measure definition
  3. Vector measure definition
  4. Hahn decomposition theorem statement
  5. Jordan decomposition of a signed measure statement
  6. Total variation of a measure definition
  7. Absolutely continuous measure definition
  8. Mutually singular measures definition
  9. Radon–Nikodym theorem statement
  10. Radon–Nikodym derivative definition
  11. Lebesgue decomposition theorem statement
  12. Measure with density definition
  13. Conditional expectation definition
  14. Vitali–Hahn–Saks theorem statement
  15. Nikodym convergence theorem statement
  16. Lyapunov's convexity theorem statement
  17. Bartle–Dunford–Schwartz integral definition

Differentiation theory 10 of 10

  1. Lebesgue differentiation theorem statement
  2. Lebesgue point definition
  3. Lebesgue's density theorem statement
  4. Vitali covering lemma statement
  5. Besicovitch covering theorem statement
  6. Hardy–Littlewood maximal function definition
  7. Doubling measure definition
  8. Fundamental theorem of calculus for Lebesgue integrals statement
  9. Integration by substitution statement
  10. Coarea formula statement

Measures on topological spaces 23 of 23

  1. Borel measure definition
  2. Regular measure definition
  3. Inner regular measure definition
  4. Outer regular measure definition
  5. Radon measure definition
  6. Tightness of measures definition
  7. Locally finite measure definition
  8. Support of a measure definition
  9. Riesz–Markov–Kakutani representation theorem statement
  10. Weak convergence of measures definition
  11. Portmanteau theorem statement
  12. Prokhorov's theorem statement
  13. Lévy–Prokhorov metric definition
  14. Wasserstein metric definition
  15. Kantorovich–Rubinstein duality statement
  16. Choquet theory concept
  17. Polish space definition
  18. Gaussian measure definition
  19. Cameron–Martin theorem statement
  20. Abstract Wiener space definition
  21. Cylinder set measure definition
  22. Minlos theorem statement
  23. Feldman–Hájek theorem statement

Measures on groups 8 of 8

  1. Haar measure definition
  2. Existence and uniqueness of Haar measure statement
  3. Modular function definition
  4. Unimodular group definition
  5. Quasi-invariant measure definition
  6. Weil's converse to Haar's theorem statement
  7. Amenable group definition
  8. Invariant mean definition

Geometric measure theory and fractal measures 18 of 19

  1. Hausdorff measure definition
  2. Hausdorff dimension definition
  3. Packing measure definition
  4. Minkowski content definition
  5. Minkowski–Bouligand dimension definition
  6. Frostman lemma statement
  7. Rectifiable set definition
  8. Besicovitch–Federer projection theorem statement
  9. Kakeya set object
  10. Kakeya conjecture conjecture
  11. Current definition
  12. Varifold definition
  13. Capacity of a set definition
  14. Choquet capacity definition
  15. Self-similar measure definition
  16. Brunn–Minkowski theorem statement
  17. Isoperimetric inequality statement
  18. Prékopa–Leindler inequality statement
  19. Hausdorff content definition

Measure-preserving maps and further topics 18 of 18

  1. Measure-preserving transformation definition
  2. Quasi-measure-preserving map definition
  3. Ergodic transformation definition
  4. Poincaré recurrence theorem statement
  5. Isomorphism theorem for measure algebras statement
  6. Standard probability space definition
  7. Standard probability space definition
  8. Kolmogorov's zero–one law statement
  9. First Borel–Cantelli lemma statement
  10. Fuzzy measure definition
  11. Choquet integral definition
  12. Young measure definition
  13. Set-valued measure definition
  14. Loeb measure definition
  15. Pettis integral definition
  16. Stieltjes function definition
  17. Hausdorff moment problem problem
  18. Hamburger moment problem problem

More ideas in this subject

Concepts 21

Theorems and results 30

Examples 6

Objects 1

Problems 1

Definitions 127

This subject is MSC class 28. See it on zbMATH Open.