Summary of contents: characteristic and simple functions; integration of non-negative, measurable, simple functions; example: integration with respect to the counting measure; basic properties of integrals of simple functions; proof that the integral against a fixed simple function defines a measure; sigma-algebra on the extended real line; real measurable functions; integration of non-negative, measurable functions; notation for integrals; Markov inequality; Monotone converge theorem and applications; proof that the integral of a non-negative measurable function is zero if and only if the function is almost everywhere zero; integral of the Dirichlet function; Lebesgue integrable functions; f^+ and f^-; integration of complex integrable functions; proof that L^1 is a vector space; Dominated convergence theorem; L^p spaces, with p in [0,infinity]; essential supremum; proof that L^2 is a complex vector space; equivalence relations, equivalence classes and quotient sets; proof that "almost everywhere equal" is an equivalence relation; Höolder's inequality; inner product on L^2.
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