**Limitations of The Riemann Integral**

Here we discuss the limitations of the Riemann integral and the greater scope offered by the Lebesgue integral. We presume a working understanding of the Riemann integral.

With the advent of Fourier series, many analytical problems involving integrals came up whose satisfactory solution required interchanging limit processes and integral signs. However, the conditions under which the integrals

- and

are equal proved quite elusive in the Riemann framework. There are some other technical difficulties with the Riemann integral. These are linked with the limit-taking difficulty discussed above.

**Failure of monotone convergence**. As shown above, the indicator function 1_{Q} on the rationals is not Riemann integrable. In particular, the Monotone convergence theorem fails. To see why, let {*a*_{k}} be an enumeration of all the rational numbers in (they are countable so this can be done.) Then let

The function *g*_{k} is zero everywhere except on a finite set of points, hence its Riemann integral is zero. The sequence *g*_{k} is also clearly non-negative and monotonically increasing to 1_{Q}, which is not Riemann integrable.

**Unsuitability for unbounded intervals**. The Riemann integral can only integrate functions on a bounded interval. It can however be extended to unbounded intervals by taking limits, so long as this doesn't yield an answer such as .

**Integrating on structures other than Euclidean space**. The Riemann integral is inextricably linked to the order structure of the line.

Read more about this topic: Lebesgue Integration

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