The benefits are powerful. The reader now has a gateway into the modern mathematics of functions. At a very early stage, undergraduates now have the required background for collaborative research in function theory.

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Large numbers of students now have significantly improved access to journal articles in analysis. The text also describes several applications of the theory, such as Fourier series, quantum mechanics, and probability.

### F.3. The Lebesgue Integral*

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The Lebesgue integral is a generalization of the Riemann integral which extends the collection of functions which are integrable. Lebesgue integration differs from Riemann integration in the way the approximations to the integral are taken. Lebesgue approximations use what are called simple functions which, like the step functions, take on only a finite number of values.

## Lebesgue integration - Wikipedia

However, these values are not necessarily taken on by the function on intervals of the domain, but rather on arbitrary subsets of the domain. The integration of simple functions under the most general circumstances possible necessitates a generalization of our notion of length of a set when the set is more complicated than a simple interval.

This report consists of the solutions of exercises found in 'Real Analysis", by H. Quotations from the book are all accompanied by the title "Definition" or "Theorem". For more details click here.

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