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))  
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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 </math>\delta<math>-fine partition exists for evey gauge </math>\delta$ is true, but is not trivial.
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The statement that a <math>\delta</math>-fine partition exists for evey gauge <math>\delta</math> is true, but is not trivial.
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==See Also==
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*[[Henstock-Kurzweil integral]]
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*[[Continuity]]
  
 
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Latest revision as of 07:53, 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 $\delta:[a,b]\rightarrow\mathbb{R}$ 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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