Difference between revisions of "Gauge"
(New page: 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 f...) |
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A funcion <math>\delta:[a,b]\rightarrow\mathbb{R}</math> is said to be a '''Gauge''' on <math>[a,b]</math> if <math>\delta(x)>0\forall x\in[a,b]</math> | A funcion <math>\delta:[a,b]\rightarrow\mathbb{R}</math> is said to be a '''Gauge''' on <math>[a,b]</math> if <math>\delta(x)>0\forall x\in[a,b]</math> | ||
| − | A [[Partition of an interval|tagged partition]] <math>\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n,</math> is said to be <i><math>\delta</math>-fine</i> on <math>[a,b]</math> if <math>[x_{i-1},x_i]\subset(t_i-\delta(t_i),t_i+\delta(t_i)) | + | A [[Partition of an interval|tagged partition]] <math>\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n,</math> is said to be <i><math>\delta</math>-fine</i> on <math>[a,b]</math> if <math>[x_{i-1},x_i]\subset(t_i-\delta(t_i),t_i+\delta(t_i))</math> |
| − | The statement that a < | + | The statement that a <math>\delta</math>-fine partition exists for evey gauge <math>\delta</math> is true, but is not trivial. |
| + | |||
| + | ==See Also== | ||
| + | *[[Henstock-Kurzweil integral]] | ||
| + | *[[Continuity]] | ||
{{stub}} | {{stub}} | ||
Latest revision as of 08:53, 16 February 2008
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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