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I had heard of the Dirichlet Function: zero at irrational points and one at rational points. It is an example of Riemann's definition of the integral not working. But I didn't know why anyone cared.

This video sketches out a sequence of Riemann integrable functions that converge to the Dirichlet Function. And makes the point: we would really like a definition of the integral that make the limit of a sequence of integrals agree with the integral of the limit of the sequence of functions.

The video doesn't get into the technical details of how the Lebesgue integral does this, but it makes it clear what motivated Henri_Lebesgue