The book covers the set of real numbers, elementary point-set topology, sequences and series of real numbers, limits and continuity, differentiation, the Riemann integral, sequences and series of functions, functions of several real variables, the

Lebesgue integral, and many other related subjects over nine chapters.

He takes a stab at it, relying heavily on the

Lebesgue integral.

Nevertheless, a proof using a basic property of the

Lebesgue integral (integral of a nonnegative-valued function is nonnegative) may be facilitated by the fact, that an infinite cone C is the intersection of all half-spaces containing it, C = [intersection] {H : H [contains] C}, where by a half space H we mean

t] and such that the

Lebesgue integral [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] along sample paths is a.

An important consequence is that an integral over G can be defined, with properties very like the usual or

Lebesgue integral.

There is also content revision in the following areas: Introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of

Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.

DELTA]] is called [DELTA]-measurable, and the

Lebesgue integral over [a, b[).

They cover measure spaces, the

Lebesgue integral, differentiation and integration, the classical Banach spaces, and extensions of additive set functions to measures.

Hubbard (Cornell University) introduces matrices for solving systems of linear equations and approximating nonlinear ones, defines smooth manifolds, discusses Riemann and

Lebesgue integrals, computes the volume of manifolds, and uses differential forms to justify the generalized Stokes's theorem.