4. Measurement theory and integration

The calculation of surface areas, i.e. the measurement of simple subsets of the plane, is as old as agriculture itself, and thus goes back to the most archaic mathematics. The parts of the plane whose areas are easiest to calculate are polygons, which can be written as a disjoint union (at the sides) of triangles and the sum of the areas of these triangles. When more complicated surfaces need to be measured, it's natural to enclose them between two "inner" and "outer" polygons whose area difference is small enough. This simple geometrical idea forms the basis of measurement theory, as developed by H. Lebesgue and his followers from 1901 onwards. We need to assign a "measure" (length, area, volume, etc.) $m_n\left(\mathrm{A}\right)$...