Measure theory was developed in successive stages during the late 19th and early 20th centuries
by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others.
The main applications of measures are in the foundations of the Lebesgue integral,
in Andrey Kolmogorov's axiomatisation of probability theory
and in ergodic theory.
Geometric measure theory (GMT) is
the study of geometric properties of sets (typically in Euclidean space) through measure theory.
It allows mathematicians to extend tools from differential geometry to
a much larger class of surfaces that are not necessarily smooth.
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