Lebesgue&#x27;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. <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Measure_%28mathematics%29." rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Measure_%28mathematics%29.</a>)