A Gauge is a more accurate way to 'measure' a partition that its norm.

A Gauge is essentially a strictly positive function defined on an interval.


A funcion $\delta:[a,b]\rightarrow\mathbb{R}$ is said to be a Gauge on $[a,b]$ if $\delta(x)>0\forall x\in[a,b]$

A tagged partition $\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n,$ is said to be $\delta$-fine on $[a,b]$ if $[x_{i-1},x_i]\subset(t_i-\delta(t_i),t_i+\delta(t_i))$

The statement that a $\delta$-fine partition exists for evey gauge $\delta$ is true, but is not trivial.

See Also

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