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The main difference between integrability in the sense of Lebesgue and Riemann is the way we measure 'the area under the curve'. The Riemann integral asks the question what's the 'height' of 𝑓f above a given part of the domain of the function. The Lebesgue integral on the other hand asks, for a given part of the range of 𝑓f, what's the measure of the 𝑥x's which contribute to this 'height'.
Integration - ELI5: Riemann-Integrable vs Lebesgue-Integrable - Mathematics Stack Exchange
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