An Introduction to Integration and Measure Theory

This book describes integration and measure theory for readers interested in analysis, engineering, and economics. It gives a systematic account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes measure from the Lebesgue-Stieltjes integral.

LIMITATIONS OF THE RIEMANN INTEGRAL. Limits of Integrals and Integrability. Expectations in Probability Theory. RIEMANN-STIELTJES INTEGRALS. Riemann-Stieltjes Integrals: Introduction. Characterization of Riemann-Stieltjes Integrability. Continuous Linear Functionals on C[a,b]. Riemann-Stieltjes Integrals: Further Properties. LEBESGUE-STIELTJES INTEGRALS. The Extension of the Riemann-Stieltjes Integral. Lebesgue-Stieltjes Integrals. MEASURE THEORY. sigma-Algebras and Algebras of Sets. Measurable Functions. Measures. Lebesgue-Stieltjes Measures. THE ABSTRACT LEBESGUE INTEGRAL. The Integral Associated with a Measure Space. The Lebesgue Spaces and Norms. Absolutely Continuous Measures. Linear Functionals on the Lebesgue Spaces. Product Measures and Fubini's Theorem. Lebesgue Integration and Measures on R?n. Signed Measures and Complex Measures. Differentiation. Convergence of Sequences of Functions. Measures on Locally Compact Spaces. Hausdorff Measures and Dimension. Lorentz Spaces. Appendices. Indexes.