Gauges and gauge-fine tagged partitions of a compact interval

Let $a\le b$. A gauge on $[a,b]$ is a function $\delta :[a,b]\to (0,\mathrm{\infty })$. A tagged partition is written

$P=\{([x_{i-1},x_i],{\xi }_i):1\le i\le m\},$

where $a=x_0<\cdots <x_m=b$ and ${\xi }_i\in [x_{i-1},x_i]$. It is $\delta$-fine when

$[x_{i-1},x_i]\subset ({\xi }_i-\delta ({\xi }_i),{\xi }_i+\delta ({\xi }_i))$

for every $i$. Its Riemann sum is $S(f,P)={\sum }_{i=1}^{m}f({\xi }_i)(x_i-x_{i-1})$.

When $a=b$, the single degenerate tagged cell $([a,a],a)$ is declared a fine tagged partition for every gauge, and its Riemann sum is $0$. A fine partial tagged partition is any finite pairwise interior-disjoint family of cells $([u_j,v_j],{\xi }_j)$ with $[u_j,v_j]\subseteq [a,b]$, $u_j\le v_j$, and ${\xi }_j\in [u_j,v_j]$, each satisfying the same gauge-containment condition. The empty family is allowed and has sum $0$.