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The Lebesgue Integral (F. Riesz’s Approach)

In: Basic Real Analysis

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  • Houshang H. Sohrab

    (Towson University, Mathematics Department)

Abstract

In this chapter we introduce a notion of integration extending the Riemann integration defined and studied in Chapter 7. The reasons for this extension are numerous and we shall not go into a detailed explanation of them. Probably the most important among them is that the space of all Riemann integrable fuctions on a compact interval [a, b] ⊂ ℝ is not complete with respect to the natural “metric”: $$ ( * ) d_1 (f,g): = \int_a^b {|f(x) - g(x)|dx (\forall f,g \in \mathcal{R}([a,b]))} . $$

Suggested Citation

  • Houshang H. Sohrab, 2003. "The Lebesgue Integral (F. Riesz’s Approach)," Springer Books, in: Basic Real Analysis, chapter 10, pages 395-430, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-8232-3_10
    DOI: 10.1007/978-0-8176-8232-3_10
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