Table of contents : 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.