Is there a strong(!) notion of integral that can face all of those issues:
- Singularities
- Decay Modes
- Oscillations
- Measure Spaces
- Locally Convex Spaces
For example combining decay modes with oscillations (non Lebesgue non Riemann): $$f:[1,\infty)\to\mathbb{R}:f(x):=\frac{1}{x}\sin(x)$$ or singularities with oscillations (non Lebesgue non Riemann): $$g:(0,1)\to\mathbb{R}:g(x):=\frac{1}{x}\sin(\frac{1}{x})$$ Moreover the famous one (non Bochner but Riemann): $$h:[0,1]\to\ell^2([0,1]):h(s):=\chi_{\{s\}}$$