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]] | ||
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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 is said to be a Gauge on if
A tagged partition is said to be -fine on if
The statement that a -fine partition exists for evey gauge is true, but is not trivial.
See Also
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