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.