Gauge
From AoPSWiki
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://alt1.artofproblemsolving.com/Forum/latexrender/pictures/7/e/7/7e70e4c3da5dd7f3e26a0ade29c5f8d35d0e2135.gif)
is said to be a Gauge on ![[a,b]](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/1/4/7/1475c39442659c6febe03b2e1809dbde6ec5fb5a.gif)
if ![\delta(x)>0\forall x\in[a,b]](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/5/0/d/50d13054d6dccbcf8c29ead762b2cfd0d21a9b7f.gif)
A tagged partition ![\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n,](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/7/6/c/76ccb02abfc163f4cc91a70202510c7224be691d.gif)
is said to be 
-fine on if ![[x_{i-1},x_i]\subset(t_i-\delta(t_i),t_i+\delta(t_i))](http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/e/2/6/e264849e99ef6b00d1ed77df0654269fd4fa191e.gif)
The statement that a -fine partition exists for evey gauge is true, but is not trivial.
See Also
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