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Measurable functions

  • Thread starter sbashrawi
  • Start date
  • #1
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Homework Statement


If E is a measurable set of measure zero, and f is bounded function on E. Is f measurable?

I tried to prove this by saying that E = { x in E | m< f(x) <M}
= {x in E | f(x) > m }intersecting { x in E | f(x) < M } and these are measurable
so f is measurable. Am I right ?


Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
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This is true as far as it goes, but you need to say specifically why the sets [tex]f^{-1}(m, \infty)[/tex] and [tex]f^{-1}(-\infty, M)[/tex] are measurable sets.
 
  • #3
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They are measurable since they are inverses of borel sets ( intervals)
 
  • #4
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In you original post: Are you talking about Borel measurability or Lebesgue measurability?
 
  • #5
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I mean lebesgue measure
 
  • #6
352
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They are measurable since they are inverses of borel sets ( intervals)
This is what you are asked to prove. You need to give a specific argument from the hypotheses why they are measurable sets.
 
  • #7
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I mean lebesgue measure
But the isn't every subset of of a set of measure zero measurable?
 

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