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Alex07
Frequent poster

Post Number: 138
Posted on Monday, 28 March, 2011 - 04:27 pm:   

Could somebody please clarify something for me?

According to my lecture notes the Lebesgue outer measure m* of a set A is defined as follows:

m*(A)=inf{ [inline]\sum\limits_{i=1}^\infty (b_i - a_i)[/inline] : [inline]A \subseteq[/inline] [inline]\bigcup_{n=1}^{\infty}(a_i, b_i)[/inline] }

According to my notes for any a>0 in the above definition we could use only intervals with [inline]b_i-a_i < a[/inline]. Why is this the case? How do we know this would not change the value of m*(A)?
lebesgue
Veteran poster

Post Number: 2089
Posted on Monday, 28 March, 2011 - 04:50 pm:   

How using long intervals could make that number smaller if longer interval can be cut to several short of the same total length? Only complication is that your definition allows open intervals only, and we cannot cut into open intervals,so we cover these long intervals with short with arbitrarily close total length. Then lnfimum in the definition will complete the job.

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