Abstract
Major convergence concepts for sequences of real-valued functions will be considered in this chapter. We have already met four convergence concepts so far (viz., pointwise, uniform, almost everywhere, and convergence in L p). These are reviewed and compared in this section. Further concepts, namely, convergence in measure, uniform almost everywhere, and almost uniform convergence, will be discussed and compared in subsequent sections.
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References
S. Abbott, Understanding Analysis, Springer, New York, 2001.
E. Asplund and L. Bungart, A First Course in Integration, Holt, Rinehart and Winston, New York, 1966.
R.G. Bartle, The Elements of Real Analysis, 2nd ed., Wiley, New York, 1976.
R.G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley, New York, 1995; enlarged 2nd ed. of The Elements of Integration, Wiley, New York, 1966.
R.G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, Vol. 32, Amer. Math. Soc., Providence, 2001.
H. Bauer, Measure and Integration Theory, Walter de Gruyter, Berlin, 2001.
S.K. Berberian, Measure and Integration, Chelsea, New York, 1965.
A. Brown and C. Pearcy, Introduction to Operator Theory I: Elements of Functional Analysis, Springer, New York, 1977.
A. Brown and C. Pearcy, An Introduction to Analysis, Springer, New York, 1995.
D.L. Cohn, Measure Theory, 2nd ed., Birkhäuser-Springer, New York, 2013.
J.B. Conway, A Course in Abstract Analysis, Graduate Studies in Mathematics, Vol. 141, Amer. Math. Soc., Providence, 2012.
J. Dugundji, Topology, Allyn & Bacon, Boston, 1966.
N. Dunford and J.T. Schwartz, Linear Operators – Part I: General Theory, Interscience, New York, 1958.
K.J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986.
K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd ed., Wiley, Chichester, 2014.
G.B. Folland, Real Analysis: Modern Techniques and their Applications, 2nd ed., Wiley, New York, 1999.
C. Goffman and G. Pedrick, First Course in Functional Analysis, Prentice-Hall, Englewood Cliffs, 1965.
P.R. Halmos, Measure Theory, Van Nostrand, New York, 1950; reprinted: Springer, New York, 1974.
P.R. Halmos, Naive Set Theory, Van Nostrand, New York, 1960; reprinted: Springer, New York, 1974.
P.R. Halmos, A Hilbert Space Problem Book, Van Nostrand, New York, 1967; 2nd ed., Springer, New York, 1982.
J.L. Kelley, General Topology, Van Nostrand, New York, 1955; reprinted: Springer, New York, 1975.
J.L. Kelley and T.P. Srinivasan, Measure and Integral – Volume 1, Springer, New York, 1988.
J.F.C. Kingman and S.J. Taylor, Introduction to Measure and Probability, Cambridge University Press, Cambridge, 1966.
A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Prentice-Hall, Englewood Cliffs, 1970.
C.S. Kubrusly, Measure Theory: A First Course, Academic Press-Elsevier, San Diego, 2007.
C.S. Kubrusly, The Elements of Operator Theory, Birkhäuser-Springer, New York, 2011; enlarged 2nd ed. of Elements of Operator Theory, Birkhäuser, Boston, 2001.
C.S. Kubrusly, Spectral Theory of Operators on Hilbert Spaces, Birkhäuser-Springer, New York, 2012.
S. Lang, Real and Functional Analysis, 3rd ed., Springer, New York, 1993.
J.N. McDonald and N.A. Weiss, A Course in Real Analysis, Academic Press, San Diego, 1999.
M.E. Munroe, Measure and Integration, 2nd ed., Addison-Wesley, Reading, 1971.
L. Nachbin, The Haar Integral, Van Nostrand, New York, 1965.
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, 2nd ed., Academic Press, New York, 1980.
F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar, New York, 1955.
C.A. Rogers, The Hausdorff Measure, Cambridge University Press, Cambridge, 1970.
H.L. Royden, Real Analysis, 3rd ed., Macmillan, New York, 1988.
W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
H. Sagan, Space-Filling Curves, Springer, New York, 1994.
G.E. Shilov and B.L. Gurevich, Integral, Measure and Derivative: A Unified Approach, Prentice-Hall, Englewood Cliffs, 1966.
P. Suppes, Axiomatic Set Theory, Van Nostrand, New York, 1960.
R.L. Vaught, Set Theory: An Introduction, 2nd ed., Birkhäuser, Boston, 1995.
A.J. Weir, Lebesgue Integration and Measure, Cambridge University Press, Cambridge, 1973.
A.J. Weir, General Integration and Measure, Cambridge University Press, Cambridge, 1974.
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Kubrusly, C.S. (2015). Convergence of Functions. In: Essentials of Measure Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-22506-7_6
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DOI: https://doi.org/10.1007/978-3-319-22506-7_6
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