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2 added hints how to define the Henstock-Kurzweil integral

The definition of the Henstock-Kurzweil integral is very similar to Riemann's integral: It is also defined using Riemann sums $\sum_{i=1}^n f(t_i) (x_i - x_{i-1})\;$, but instead of letting $\delta = \mathop{max} (x_i - x_{i-1})\;$ approach $0$, one considers for gauge functions $\delta : [a, b] \to \mathbb{R}_{>0}\;$ compatible tagged partitions $(t_i, [x_{i-1}, x_i])_i\;$ with $t_i - \delta(t_i)< x_{i-1}\le t_i \le x_i < t_i +\delta(t_i)\;\;$ and then defines the integral in essential the same way (for details see the wikipedia article).

Dropping the condition $t_i \in [x_{i-1}, x_i]$ one gets an integral (McShane integral) equivalent to the Lebesgue integral on the line, whereas the Henstock-Kurzweil integral works also for "non-absolutely convergent" cases like $\int_0^1 \frac{\mathop{sin}(1/x)}{x} dx\;$.

As one gets all the nice theorems (monotone convergence theorem, dominated convergence theorem, second fundamental theorem of calculus, Lebesgue differentiation theorem) for the cost of a slightly (?) more difficult definition, there shouldn't be any reason (besides tradition) for teaching Riemann integration on $\mathbb R$ to mathematically moderately mature students (and traditions can be satisfied by mentioning that restricting to constant gauge functions gives the Riemann integral).

For more than one dimension (maybe in a second course), one can/should/must then teach Lebesgue integration. Here I have to add from my own experience (I got introduced to integrals equivalent to Lebesgue's in at least five different courses) that a clean separation between measure theory and topology at the beginning like it is done e.g. in Heinz Bauer's "Measure and integration theory" helps enormously for a deeper understanding.

1 [made Community Wiki]

hilbertthm90 and Maxime Bourrigan mentioned already in the comments to the question that the Henstock-Kurzweil integral offers a good alternative to the Riemann integral (see also the lecture notes referred to by Maxime) for a course offered for mathematicians.