Given a compact space $\Omega$ and a Banach space $E$
Consider functions $f:\Omega\to E$.
Regard neighborhood gauges: $$\delta:\Omega\to\mathcal{T}(\Omega):\quad\delta(\omega)\in\mathcal{N}(\omega)$$ and finite measurable tagged partitions: $$\mathcal{P}^*\subseteq\mathcal{B}(\Omega):\quad\#\mathcal{P}^*<\infty$$ (In fact, the tags are just surpressed.)
Order gauges by inclusion: $$\delta\leq\delta':\iff\delta(\omega)\subseteq\delta'(\omega)\quad(\omega\in\Omega)$$ and collect gauge-fine partitions: $$\mathcal{P}^*\dashv\delta:\iff A_n\subseteq\delta(a_n)\quad(A_n\in\mathcal{P}^*)$$
Denote the partial sums by: $$\mathcal{S}(\mathcal{P}^*)=\sum_nF(a_n)\lambda(A_n)$$ and define the gauge integral to be the limit: $$\int_\Omega F\mathrm{d}\lambda:=\lim_\delta\{\mathcal{S}(\mathcal{P}^*)\}_{\mathcal{P}^*\dashv\delta}$$
Why is the value assigned to a gauge integral well defined (unique)?
Gauges are directed: $$\delta\wedge\delta'(\omega):=\delta(\omega)\cap\delta'(\omega)$$ and every gauge admits a partition: $$\Omega=\bigcup_{n=1}^N\delta(a_n)=:\bigcup_{n=1}^NA_n:\quad\mathcal{P}_\delta:=\{A_1',\ldots,A_N'\}\quad(A_n':=A_n\setminus\cup_{m=1}^{n-1}A_m)$$ (Note that compactness was necessary!)
That is gauges induce a net of partial sums: $$\mathcal{S}(\mathcal{P}_\delta^*):\quad{\mathcal{P}}_{\delta}^{*}\leq{\mathcal{P}'}_{\delta'}^{*'}:\iff\delta\leq\delta'$$ (Caution: There may be no meet!)
Concluding that the integral is well-defined in Banach space $E$.