==Lebesgue Integral==
The Lebesgue integral is usually introduced in late university or early postgraduate mathematics. It is naively described as rotating the Reimann integral, in that it is the range instead of the domain that is partitioned. An understanding of [[measure theory]] is required to understand this technique.
The Lebesgue integral is defined for every function for which the Riemann integral is defined, as well as for an even larger class of functions: this possibility of integrating functions which are not Riemann integrable is a large part of the motivation for the Lebesgue theory. A typical example of a function which is Lebesgue integrable but not Riemann integrable is the [[characteristic function]] of the [[rational number|rationals]], <math>\chi_{\mathbb Q}</math>, defined by
::<math>\chi_{\mathbb Q}(x) =
\begin{cases}
1 & \mbox{if } x \in \mathbb Q, \\
0 & \mbox{if } x \notin Q
\end{cases}</math>.
Because the rationals are only countable, this function is zero "almost everywhere": there are far more irrationals than rationals, and the function is <math>0</math> at all of these. Thus we would expect that
::<math>\int_0^1 \chi_{\mathbb Q}(x) \, dx = 0</math>
Unfortunately, this function has so many discontinuities that its Riemann integral is not defined. However, if we use the Lebesgue integral instead, the function is integrable as hoped, and has the expected value <math>0</math>.
==See Also==