Gauge
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.
Definition
A funcion ![$\delta:[a,b]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/d/0/6/d06f3bfaeb41926ee6a1165d3fa1863b35ce1f5d.png)
is said to be a Gauge on ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
if ![$\delta(x)>0\forall x\in[a,b]$](http://latex.artofproblemsolving.com/4/d/0/4d0711cd8a707dc07303b0c5bc2e0a6164baf955.png)
A tagged partition ![$\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n,$](http://latex.artofproblemsolving.com/b/7/e/b7e611d201b87739261e63095fdc63b6e0ca6fac.png)
is said to be 
-fine on if ![$[x_{i-1},x_i]\subset(t_i-\delta(t_i),t_i+\delta(t_i))$](http://latex.artofproblemsolving.com/2/e/a/2ea6ec504d6ed3dfed294b42d9e222742483ad4a.png)
The statement that a -fine partition exists for evey gauge is true, but is not trivial.
See Also
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