Robert G. Bartle 
The Elements of Integration and Lebesgue Measure 

Consists of two separate but closely related parts. Originally published in 1966, the first section deals with elements of integration and has been updated and corrected. The latter half details the main concepts of Lebesgue measure and uses the abstract measure space approach of the Lebesgue integral because it strikes directly at the most important results–the convergence theorems.

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THE ELEMENTS OF INTEGRATION.

Measurable Functions.

Measures.

The Integral.

Integrable Functions.

The Lebesgue Spaces Lp.

Modes of Convergence.

Decomposition of Measures.

Generation of Measures.

Product Measures.

THE ELEMENTS OF LEBESGUE MEASURE.

Volumes of Cells and Intervals.

The Outer Measure.

Measurable Sets.

Examples of Measurable Sets.

Approximation of Measurable Sets.

Additivity and Nonadditivity.

Nonmeasurable and Non-Borel Sets.

References.

Index.

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Robert Gardner Bartle was an American mathematician specializing in real analysis. He is known for writing various popular textbooks.