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It seems that measure theory has a very good theoretical purpose, in that it provides a rigorous framework to define a lot of what we do in analysis. However, I have a hard time thinking of a situation where you need to invoke a purely measure-theoretic concept that is not serving as a "shoring up" lemma/theorem to the "main idea", which will often be some integral or limit.

Are there instances where the measure theoretic idea is the crux of a result that isn't related to probability theory? At the broader level, where are non-probabilistic measureable spaces even used?

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    $\begingroup$ en.wikipedia.org/wiki/Geometric_measure_theory $\endgroup$ – Will Jagy Dec 14 '14 at 20:15
  • $\begingroup$ @WillJagy Thanks, that looks like a good example of what I was talking about. Diff. Geo + measure theory > Diff. Geo :) $\endgroup$ – user76844 Dec 14 '14 at 20:44
  • $\begingroup$ What's a non probabilistic measure space? One where the measure isn't real, nonnegative, and bounded? $\endgroup$ – Tim kinsella Dec 14 '14 at 21:02
  • $\begingroup$ @Tim Kinsella yes...any measurable space not satisfying conditions for a probability space $\endgroup$ – user76844 Dec 14 '14 at 22:06
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@WillJagy pointed me to Geometric Measure Theory, which requires measures to understand areas/lengths of things like fractals.

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