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It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first decades of the 20th century. [1] [2] A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will have ...
Burnside's lemma also known as the Cauchy–Frobenius lemma; Frattini's lemma (finite groups) Goursat's lemma; Mautner's lemma (representation theory) Ping-pong lemma (geometric group theory) Schreier's subgroup lemma; Schur's lemma (representation theory) Zassenhaus lemma
Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event is expected to occur approximately equals the probability of the event's occurrence on any particular trial; the larger the ...
Proofs of Borel's lemma can be found in many text books on analysis, including Golubitsky & Guillemin (1974) and Hörmander (1990), from which the proof below is taken. Note that it suffices to prove the result for a small interval I = (− ε , ε ), since if ψ ( t ) is a smooth bump function with compact support in (− ε , ε ) equal ...
The concept of a normal number was introduced by Émile Borel . Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers. Wacław Sierpiński showed that it is possible to specify a particular such number.
Borel algebra; Borel measure; Indicator function; Lebesgue measure; Lebesgue integration; Lebesgue's density theorem; Counting measure; Complete measure; Haar measure; Outer measure; Borel regular measure; Radon measure; Measurable function; Null set, negligible set; Almost everywhere, conull set; Lp space; Borel–Cantelli lemma; Lebesgue's ...
The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set X. Using the discrete metric. The Borel–Cantelli lemma is an example application of these constructs. Using either the discrete metric or the Euclidean metric
Borel–Cantelli lemma; C. Covering lemma; ... Vitali covering lemma; W. Whitney covering lemma This page was last edited on 1 January 2018, at 13:47 (UTC) ...