Very shortly. Consider for instance the Behrens-FisherBehrens–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) and, more generally, point (i.e. Lebesgue negligible) null hypothesis testing problems.
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 on uncountable sets, for the probability that the numerical values of two continuous parameters with absolutely continuous marginal probability measures wrt the Lebesgue measure on $\mathbb{R}$ are equal to each other is equal to zero, a priori and a posteriori. Therefore, the Behrens-FisherBehrens–Fisher problem with continuous parameters admits a trivial but totally useless and meaningless solution under measure theory on uncountable sets.
That's the reason why the standard Bayesian solution to this problem, found in any textbook, is plain wrong. In particular, it violates measure theory on uncountable sets, by assigning non-zero probabilities to Lebesgue-negligible (e.g. singletons) sets. See A fully Bayesian solution to k-sample tests for comparison and the Behrens-FisherBehrens–Fisher problem based on the Henstock-KurzweilHenstock–Kurzweil integral..
To get the correct, meaningful, useful and practical solution to this problem, we have to forget measure theory on uncountable sets, that is we have to forget actual, uncountable infinity and go back to potential infinity and measure theory on finite sets, in order to take the limit of the solutions to the discrete problems. By construction, this limit solution, the Bayes-PoincaréBayes–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]$$$\left.\left[ {\frac{{\int\limits_{{\Theta _0}} {\prod\limits_{i = 1}^k {{p_{\left. {{\theta _i}} \right|{x_i}}}\left( \theta \right)} {\text{d}}\theta \,} }}{{\prod\limits_{i = 1}^k {\int\limits_{{\Theta _i}} {{p_{\left. {{\theta _i}} \right|{x_i}}}\left( \theta \right){\text{d}}\theta \,} } }}} \right]\middle/\left[ {\frac{{\int\limits_{{\Theta _0}} {\prod\limits_{i = 1}^k {{p_{{\theta _i}}}\left( \theta \right)} {\text{d}}\theta \,} }}{{\prod\limits_{i = 1}^k {\int\limits_{{\Theta _i}} {{p_{{\theta _i}}}\left( \theta \right){\text{d}}\theta \,} } }}} \right]\right.$$
is obtained from the limits of Riemann sums that are, by definition, Cauchy-Riemann-Darboux-...-Denjoy-Luzin-Perron-Henstock-KurzweilCauchy–Riemann–Darboux–…–Denjoy–Luzin–Perron–Henstock–Kurzweil integrals but not Lebesgue's, since we had to forget measure theory on uncountable sets for a while. Conversely, I consider that the Behrens-FisherBehrens–Fisher problem remained unsolved for such a long time (until there is evidence to the contrary) due to the old habit of jumping directly into the actual, uncountable infinity within the standard Borel-Lebesgue-KolmogorovBorel–Lebesgue–Kolmogorov measure-theoretic setting for probability theory.
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 PoincaréLy - Identité et égalité, le criticisme de Poincaré:
That's the most fundamental reason why I believe one should still teach Riemann integration or Henstock-Kurzweil'sHenstock–Kurzweil's like in Belgium, as soon as one has some physical, statistical or practical applications in mind, because we cannot always jump directly into actual infinity.
I don't agree with Dieudonné: the Henstock-KurzweilHenstock–Kurzweil integral is another sensible answer, there are pros and cons for both integrals, depending on one's point of view.
In his Eléments d'histoire des mathématiques, Bourbaki mentions the "deep works" of Denjoy, Perron, de la Vallée-Poussin, Khintchine, LusinLuzin, Banach, etc. (chapter Intégration, p. 282, third edition, 1974) but not Kurzweil's and Henstock's.
