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Measure theory and Cantor function

  • Thread starter sbashrawi
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Homework Statement


Show that there is a continuous , strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero


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The Attempt at a Solution



I need to find a mapping to a countable set or cantor set but I couldn't construct such function
 

Answers and Replies

  • #2
Dick
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TRY the Cantor function. Why doesn't it work? What might you do to modify it?
 
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Cantor function is amapping from [0,1] onto [0,1] , I need a restriction so that its image has measure zero. But it is seemed to me that its image is countable am I right?
 
  • #4
Dick
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Cantor function is amapping from [0,1] onto [0,1] , I need a restriction so that its image has measure zero. But it is seemed to me that its image is countable am I right?
It's image is [0,1]. That's not countable. Nor does it have measure zero. But those aren't the things you need. If C(x) is the Cantor function, it's not strictly monotone increasing, is it? Can you tell me why not? There's an easy way to modify it into a function from [0,1] to [0,2] that is strictly monotone increasing.
 
  • #5
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We have the function : h(x) = C(x) + x , where C(x) is the cantor function, the function h(x) is increasing function from [0,1] onto [0,2].
[0,1] = C union O : c = cantor set , O its complement.
m( C) = 0 , m(O) = 1 .
how can I modify this to get the required function?
 
  • #6
Dick
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We have the function : h(x) = C(x) + x , where C(x) is the cantor function, the function h(x) is increasing function from [0,1] onto [0,2].
[0,1] = C union O : c = cantor set , O its complement.
m( C) = 0 , m(O) = 1 .
how can I modify this to get the required function?
That's already (almost) the function you want. m([0,2])=2. What are m(h(O)) and m(h(C))?
 
  • #7
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m( h(O)) = 1 and m(h(c))= 1 .
I think we can use the inverse function of h . it is contniuous , increasing
and hinv: [0,2] to [0,1]
and h inv(h(c)) = c which has measure zero i.e it maps a set with positive measure to a set of measure zero as required.

let f(x) = h(x) / 2 we get the required function: f: [0,1] to [0,1]
 
  • #8
Dick
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m( h(O)) = 1 and m(h(c))= 1 .
I think we can use the inverse function of h . it is contniuous , increasing
and hinv: [0,2] to [0,1]
and h inv(h(c)) = c which has measure zero i.e it maps a set with positive measure to a set of measure zero as required.

let f(x) = h(x) / 2 we get the required function: f: [0,1] to [0,1]
That's it.
 

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