The Lebesgue-Radon-Nikodym theorem and how von Neumann proved it

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An introductionIf one wants to learn the fundamental theorem of Calculus in the sense of Lebesgue integral, properties of measures have to be taken into account. In elementary calculus, one may consider something likedf(x)=f'(x)dxwhere $f$ is differentiable, say, everywhere on an interval. Now we re……