monotone convergence theorem (on a monotone pointwise limit) does not hold for Riemann integrals. Thus, in Riemann integration, taking limits under the integral sign is far more difficult to logically justify than in Lebesgue integration. Generalizations It is easy to extend the Riemann integral to functions with values in the Euclidean vector space for any . The integral is defined component-wise; in other words, if then In particular, since the complex numbers are a real vector space, this allows the integration of complex valued functions. The Riemann integral is only defined on bounded intervals, and it does not extend well to unbounded intervals. The simplest possible extension is to define such an integral as a limit, in other words, as an improper integral: This definition carries with it some subtleties, such as the fact that it is not always equivalent to compute the Cauchy principal value For example, consider the sign function which is 0 at , 1 for , and −1 for . By symmetry, always, regardless of . But there are many ways for the interval of integration to expand to fill the real line, and other ways can produce different results; in other words, the multivariate limit does not always exist. We can compute In general, this improper Riemann integral is undefined. Even standardizing a way for the interval to approach the real line does not work because it leads to disturbingly counterintuitive results. If we agree (for instance) that the improper integral should always be then the integral of the translation is −2, so this definition is not invariant under shifts, a highly undesirable property. In fact, not only does this function not have an improper Riemann integral, its Lebesgue integral is also undefined (it equals ). Unfortunately, the improper Riemann integral is not powerful enough. The most severe problem is that there are no widely applicable theorems for commuting improper Riemann integrals with limits of functions. In applications such as Fourier series it is important to be able to approximate the integral of a function using integrals of approximations to the function. For proper Riemann integrals, a standard theorem states that if is a sequence of functions that converge uniformly to on a compact set , then On non-compact intervals such as the real line, this is false. For example, take to be on and zero elsewhere. For all we have: The sequence converges uniformly