The Lebesgue Integral (F. Riesz’s Approach)

Houshang H. Sohrab
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Houshang H. Sohrab: Towson University, Mathematics Department

Chapter 10 in Basic Real Analysis, 2003, pp 395-430 from Springer

Abstract: 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”: $$ ( * ) d_1 (f,g): = \int_a^b {|f(x) - g(x)|dx (\forall f,g \in \mathcal{R}([a,b]))} . $$

Date: 2003
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DOI: 10.1007/978-0-8176-8232-3_10

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