If the students are supposed to do only pure mathematics later on in their professional life, then I have nothing to say.

But if they are ever supposed to do some applied mathematics, physics or statistics, then my humble opinion is that you should never teach them only the Lebesgue integral.

For a simple and sufficient reason:

**Measure theory itself disqualifies the Lebesgue integral (on which it relies) at least for some important problems of (applied) probability theory.** 

Very shortly. Consider the Behrens-Fisher problem, the most famous 90 years old open problem in statistics and applied probability theory, with vital applications such as clinical trials (e.g. placebo vs treatment).

If this problem is trivial for discrete parameters of interest, it does not make sense for continuous ones from the point of view of Bayesian probability theory + measure theory, for the probability that the numerical values of two continuous parameters are equal to each other is equal to zero, a priori and a posteriori.

This is the reason why the standard Bayesian solution to this problem, found in any textbook, is plain wrong. In particular, it violates measure theory. See [A fully Bayesian solution to k-sample tests for comparison and the Behrens-Fisher problem based on the Henstock-Kurzweil integral.][3]

In order to get the correct solution, we have to forget actual infinity and go back to potential infinity to take the limit of the solutions to the discrete problems. By construction, this limit solution, the Bayes-Poincaré factor 

$\left[ {\frac{{\int\limits_{{\Theta _0}} {\prod\limits_{i = 1}^k {{p_{\left. {{\theta _i}} \right|{x_i}}}\left( \theta  \right)} {\text{d}}\theta \,} }}{{\prod\limits_{i = 1}^k {\int\limits_{{\Theta _i}} {{p_{\left. {{\theta _i}} \right|{x_i}}}\left( \theta  \right){\text{d}}\theta \,} } }}} \right]/\left[ {\frac{{\int\limits_{{\Theta _0}} {\prod\limits_{i = 1}^k {{p_{{\theta _i}}}\left( \theta  \right)} {\text{d}}\theta \,} }}{{\prod\limits_{i = 1}^k {\int\limits_{{\Theta _i}} {{p_{{\theta _i}}}\left( \theta  \right){\text{d}}\theta \,} } }}} \right]$

is obtained from the limits of Riemann sums that are, by definition, Cauchy-Riemann-Darboux-...-Denjoy-Luzin-Perron-Henstock-Kurzweil integrals but not Lebesgue's, as far as I can understand. Conversely, I consider that the Behrens-Fisher problem remained unsolved for such a long time due to the supremacy of the Lebesgue integral.  

This is in line with Poincaré's considerations about the physical continuum versus the mathematical one, see for instance [Identité et égalité, le criticisme de Poincaré][1]:

> If, indeed, Poincaré defends the philosophical thesis according to
> which we must understand the continuum as a potential infinity, it is
> not for lack of questioning it, but because it finds in potential
> infinity a virtue that actual infinity does not possess. It is
> therefore not only because of the paradoxes that it engenders that
> Poincaré rejects actual infinity, it is because by considering the
> infinity of the continuum as actual and not as potential, we do not
> see how it gives mathematics the specificity that makes it the
> privileged language of physics. For then one is forbidden to take into
> account both the sensible intuition and the intuition a priori that
> preside over the mathematical conception of the continuum and whose
> importance is only understood once the positive virtue of the
> potential infinity is recognized.

and Poincaré's silence on the Lebesgue integral and measure theory between 1904 and 1912 is no surprise [An historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?][2].

That's the reason why I belive one should still teach Riemann integration or Henstock-Kurzweil's like in Belgium, as soon as one has some physical, applied or practical applications in mind.

[1]: https://mathoverflow.net/questions/345275/an-historical-mystery-poincar%c3%a9-s-silence-on-lebesgue-integral-and-measure-theo#345284
[2]: https://mathoverflow.net/questions/345275/an-historical-mystery-poincar%c3%a9-s-silence-on-lebesgue-integral-and-measure-theo#345284
[3]: https://www.researchgate.net/publication/335859251_A_fully_Bayesian_solution_to_k-sample_tests_for_comparison_and_the_Behrens-Fisher_problem_based_on_the_Henstock-Kurzweil_integral