Abstract
This chapter examines the history of measure theory. At the beginning of civilization, mathematics could be differentiated from science and technology as rational art of solving abstract problems with numbers and geometrical figures. It means that the solution has to be obtained from initial data with some kind of rational, logical mental process. Greek philosophers and mathematicians made the great discovery that mathematical rules could be proved and that the whole of mathematics can be organized in axiomatic theory. Pappos proves that the spiral and quarter of the circumference that connects its starting and end point bound the surface, which has the same ratio to the hemisphere as the sector of the quadrant to the quadrant. Translational invariance of Lebesgue integral has generalization with important application in the representation theory of groups. It is suggested that much of character and representation theory of finite groups remains valid if suitable integration is applied over the compact manifold formed by the elements of the rotation group.
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