ABSTRACT
A substantial amount of integration theory of functions of one variable carries over to the case of functions defined on a subset of Rn. Nevertheless, there are some important differences, and the proper theory of multiple integrals was fully developed only at the end of the 19th century, in the work of Thomae, Peano, and Jordan. Much of the effort focused on the attempts to “measure” sets in R2. This laid the foundation for the work of Borel and Lebesgue in the 20th century, which resulted in the modern theory of measure.
