differences between Riemann and Lebesgue integral

Both the Riemann integral and the Lebesgue integral do the same thing. They find the area under a function. They main (and really only) difference is in how they partition the domain to approximate the area.

The Riemann integral partitions along the domain into finite intervals then uses a point along the function as the approximation.

The Lebesgue integral partitions along the codomain along finite values and finds the measure of the inverse image of these sections. Because it uses the measure of the domain and the machinery of the inverse image, the Lebesgue integral can integrate “finer” functions. In an informal way, it can get into smaller places than the intervals of the Riemann integral.

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not Lebesgue integrable function not Riemann integrable function