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Article 44

Convergence Theorems

Real Analysis Exchange 25 (1999/2000), no. 2, 955-958.

Mathematical reviews subject  classification: 26A30.

 

Abstract

This is a query, and we ask if the following classical theorems on convergence are equivalent: 

  1. the Lebesgue-Beppo Levi Theorem (see N1, p. 141)
  2. a theorem on the integration of a series with positive terms (see N1, p. 142)
  3. the Fatou Lemma I  (see H1, p. 172)
  4. the Fatou Lemma II  (see N1, p. 140)
  5. the Fatou Lemma III (see N1, p. 140) 
  6. Lebesgue's Dominated Convergence Theorem I (see H1, p. 172)
  7. Lebesgue's Dominated Convergence Theorem II (see H1, p. 173)
  8. Lebesgue's Dominated Convergence Theorem III (see N1, pp. 149-50)
  9. Vitali's Theorem (see N1, p. 152)
  10. Lebesgue's Dominated Convergence Theorem for Bounded Functions I (see N1, p. 127)
  11. Lebesgue's Dominated Convergence Theorem for Bounded Functions II.

 

Reference:

H1: E. Hewitt and K. Stromberg, Real and abstract analysis, Springer Verlag, 1969.

N1: I. P. Natanson, Theory of functions of a real variable, 2nd. rev. ed., Ungar, New York, 1961.