Reference
For a bounded nonexample of integrability see: Riemann Integral: Bounded Nonexample
For a convergence theorem on integral see: Riemann Integral: Uniform Convergence
For a comparison of integrals see: Uniform Integral vs. Riemann Integral
Definition
Given a measure space $\Omega$ and a Banach space $E$.
Consider functions $F:\Omega\to E$.
Denote the measurable subsets of finite mass by: $$\mathcal{A}_\infty:=\{A:\mu(A)<\infty\}$$ and order them by inclusion: $$A\leq A':\iff A\subseteq A'$$
Remember the generalized Riemann integral on finite measure spaces: $$A\in\mathcal{A}_\infty:\quad\int_AF\mathrm{d}\mu:=\lim_\mathcal{P}\left\{\sum_{a\in A\in\mathcal{P}}F(a)\mu(A)\right\}_\mathcal{P}$$ (For more details see references above.)
Define the improper Riemann integral as: $$\int_\Omega F\mathrm{d}\mu:=\lim_A\left\{\int_AF\mathrm{d}\mu\right\}_{A\in\mathcal{A}_\infty}$$ (Crucially, this reflects independence of approximation by finite spaces.)
Discussion
For finite measure spaces the improper agrees with the proper as $\Omega\in\mathcal{A}_\infty$.
This way, poles still can't be handled: $$\int_0^1\frac{1}{\sqrt{x}}\mathrm{d}x\notin E$$ (Note that the concept of compact intervals isn't available in general.)
For Borel spaces a suitable criterion could be continuity plus absolute integrability: $$F\in\mathcal{C}(\Omega,E):\quad\int_\Omega\|F\|\mathrm{d}\mu<\infty\implies\int_\Omega F\mathrm{d}\mu\in E$$
How to prove this in the abstract setting?
(I slightly doubt it...)
