Lebesgue's theory of integration is built on measure theory, which was developed by Lebesgue and others. Measure theory has turned out to be a very natural fra

Lebesgue's theory of integration is built on measure theory, which was developed by Lebesgue and others. Measure theory has turned out to be a very natural framework for probability, stochastic processes, ergodic theory, etc. (See e.g. https://en.wikipedia.org/wiki/Measure_%28mathematics%29. https://en.wikipedia.org/wiki/Measure_%28mathematics%29.)