A function is Riemann integrable if it can be approximated by a sum of areas of rectangles under its curve, allowing us to compute the total area under that curve over a specified interval. This concept connects to Lebesgue measure and integration by highlighting the limitations of Riemann integration in dealing with more complex functions, particularly those with discontinuities or irregular behavior. While Riemann integrability is sufficient for many functions encountered in calculus, Lebesgue integration extends the notion to a broader class of functions, making it a crucial tool in advanced mathematical analysis.
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