In this thesis we look at the Henstock-Kurzweil integral. Its definition is similar to Riemann’s, except that the fineness of the partition of intervals is determined by a function $\delta$ and no longer by a constant. This difference allows one to integrate much more general functions. Leibniz’s formula allows the integration of functions which are not necessarily defined everywhere on a closed interval. This is followed by the introduction of improper integrals, which do not extend the set of integrable functions at all. The main result is the monotone convergence theorem, which gives a very manageable characterisation of function sequences for which the order of limit and integration can be reversed.
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