Editing Uniform convergence (section)

===To integrability===
Similarly, one often wants to exchange integrals and limit processes. For the [[Riemann integral]], this can be done if uniform convergence is assumed:
: ''If <math>(f_n)_{n=1}^\infty</math> is a sequence of Riemann integrable functions defined on a [[compact space|compact]] interval <math>I</math>  which uniformly converge with limit <math> f</math>, then <math> f</math> is Riemann integrable and its integral can be computed as the limit of the integrals of the <math> f_n</math>:'' <math display="block">\int_I f = \lim_{n\to\infty}\int_I f_n.</math>
In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function.  This follows because, for ''n'' sufficiently large, the graph of <math>f_n</math> is within {{math|&epsilon;}} of the graph of ''f'', and so the upper sum and lower sum of <math>f_n</math> are each within <math>\varepsilon |I|</math> of the value of the upper and lower sums of <math>f</math>, respectively.

Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the [[Lebesgue integration|Lebesgue integral]] instead.