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Notes on the Lebesgue Integral

In the definition of the Riemann integral of a function f(x), the x-axis is partitioned and the integral is defined in terms of limits of the Riemann sums ∑n−1 j=0 f(x ∗ j )∆j , where ∆j = xj+1−xj . The basic idea for the Lebesgue integral is to partition the y-axis, which contains the range of f , rather than the x-axis. This seems like a “dumb” idea at first. Shouldn’t the two ways end up giving the same integral? Most of time this is the case, but Lebesgue was after integrating some functions for which the Riemann integral doesn’t exist; for example, the Dirichlet function, which is defined on [0, 1]: