Disclaimer
This thread is meant to record. See: Answer own Question
Anyway, it is written as problem. Have fun! :)
Reference
This thread is directly related to:
- Bochner Integral: Axioms
- Bochner Integral: Integrability
- Riemann Integral: Bounded Nonexample
- Riemann Integral: Uniform Convergence
- Riemann Integral: Improper Version
- Riemann Integral vs. Uniform Integral
and it is indirectly related to:
Prologue
This is the final comparison.
All notions are up to null sets.
Definitions
Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$.
Consider plain functions $F:\Omega\to E$.
Denote the Bochner measurable functions by $\mathcal{B}$.
Define the Bochner integral by: $$F\in\mathcal{B}:\quad\int F\mathrm{d}\mu:=\lim_n\int S_n\mathrm{d}\mu$$ for simple functions satisfying: $$\int\|F-S_n\|\mathrm{d}\mu\to0$$ Denote the Bochner integrable functions by $\mathcal{L}_\mathfrak{B}$.
Define the generalized Riemann integral by: $$\int_\mathfrak{R}F\mathrm{d}\mu:=\lim_\mathcal{P}\{\sum_{a\in A\in\mathcal{P}}F(a)\mu(A)\}_\mathcal{P}$$ over finite measurable partitions: $$\mathcal{P}\subseteq\Sigma:\quad\Omega=\bigsqcup_{A\in\mathcal{P}}A\quad(\#\mathcal{P}<\infty)$$ being ordered by refinement: $$\mathcal{P}\leq\mathcal{P}':\iff\forall A'\in\mathcal{P}'\exists A\in\mathcal{P}:\quad A\supseteq A'$$ Denote Riemann integrable functions by $\mathcal{L}_\mathfrak{R}$.
Problem
Clearly, boundedness is a necessary condition for Riemann integrability: $$f:(0,1]\to\mathbb{R}:x\mapsto\frac{1}{\sqrt{x}}:\quad f\in\mathcal{L}_\mathfrak{B},\,f\notin\mathcal{L}_\mathfrak{R}$$
However, there are also bounded Riemann integrable functions that are not Bochner integrable: $$F:[0,1]\to\ell^2[0,1]:t\mapsto\chi_t:\quad F\notin\mathcal{L}_\mathfrak{B},\,F\in\mathcal{L}_\mathfrak{R}$$
Moreover, both strictly include the uniform integral: $$\mathcal{L}_\mathfrak{U}\subsetneq\mathcal{L}_\mathfrak{B}\land\mathcal{L}_\mathfrak{U}\subsetneq\mathcal{L}_\mathfrak{R}$$
So the remaining question is wether at least for the bounded case: $$\|F\|_\infty<\infty:\quad F\in\mathcal{L}_\mathfrak{B}\implies F\in\mathcal{L}_\mathfrak{R}$$ and especially: $$\int_\mathfrak{R}F\mathrm{d}\mu=\int_\mathfrak{B}F\mathrm{d}\mu$$ (Have a guess! ;))