Measure theory

1 artículo

Lebesgue measure Measurable function Lebesgue integration Lp space Radon nikodym Measure Measure space Measurable space Non measurable set Vitali set Nikodym set Outer measure Hausdorff density Lebesgue integral Lebesgue stieltjes integral Daniell integral Bochner integral Pettis integral Choquet integral Kolmogorov integral Khinchin integral Burkill integral Pfeffer integral Hellinger integral Direct integral Dominated convergence theorem Fatou's lemma Convergence in measure Almost everywhere Essential infimum and essential supremum Caratheodory's criterion Lebesgue point Layer cake representation Distribution function Cylindrical sigma algebra Standard probability space Conull set Tightness of measures Vague topology Levy prokhorov metric Wasserstein metric Levy metric Portmanteau theorem Vector measure Positive and negative set Sigma additive set function Measure algebra Information theory and measure theory Measure preserving dynamical system Radon nikodym set Regular conditional probability Differentiable measure Finite dimensional distribution Hopf decomposition Lifting theory Essential range Prevalent and shy set Strong measure zero set Klee's measure problem Intensity Tau additivity Bochner measurable function Weakly measurable function Pettis theorem Absolutely continuous invariant measure Absolutely continuous measures Absolutely integrable function Algebraic independence measure of Analytic borel space Approximation of functions measure of Bernoulli measure Borel function Borel isomorphism Borel lebesgue covering theorem Borel measure Borel strong law of large numbers Borel summation method Borel system of sets Borel transform Carath odory measure Carleson measure Complete measure Completely integrable differential equation Conformal measure Contiguity of probability measures Cylindrical measure Differentiation of measures Discrete measure Energy of measures Favard measure Feynman measure Fourier borel transform Harmonic measure Harmonic measure principle of Kullback leibler type distance measures Lebesgue constant Lebesgue constants of multi dimensional partial fourier sums Lebesgue criterion Lebesgue decomposition Lebesgue dimension Lebesgue function Lebesgue inequality Lebesgue number Lebesgue space Lebesgue spectrum Lebesgue summation method Lebesgue theorem Linear independence measure of Mackey borel structure Measure of association Measure preserving transformation Non additive measure Optional sigma algebra Perfect measure Pre measure Predictable sigma algebra Radon integral Radon nikod m theorem Repeated measures Signed measure Singular measures Spectral measure Support of a measure Tamagawa measure Term generated sigma algebra Tight measure Transcendency measure of Vector valued measure Weak convergence of probability measures Wiener measure Young measure

Analysis / Calculus GradoAnálisis14 min

Every Upper Sum Says One, Every Lower Sum Says Zero, and the Gap Is the Measure of the Discontinuities

On the unit interval the indicator of the rationals has upper sum 1 and lower sum 0 for every partition ever written, so the two never meet and Riemann returns nothing at all. Give the k-th rational an interval of width ε/2^k and the whole countable set sits inside a total length of ε, which puts its measure at 0 and its Lebesgue integral at 0. Lebesgue's criterion turns that into the general law — a bounded function on a compact interval is Riemann integrable exactly when its discontinuities have measure zero — which is why Thomae's function, discontinuous on the same dense set, is integrable and this one is not. The trade is not free: sin x over x on the half line has an improper Riemann value of π/2 and no Lebesgue integral at all.