Proving Measure Zero for Set A Derived from Real Numbers Set E

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Discussion Overview

The discussion centers on proving that a set A, derived from a set E of real numbers, has measure zero. The context involves theoretical exploration of measure theory and the properties of sets in relation to measure zero.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests approximating the set E using open intervals, proposing that if E is contained within the union of these intervals, the sum of their lengths can be made arbitrarily small.
  • Another participant raises a concern regarding the boundedness of intervals, questioning the validity of squaring the intervals to approximate set A.
  • A different participant reiterates the issue of boundedness, emphasizing that while any bounded set of measure zero can be split into countably many subsets, the approach may not hold for unbounded sets.

Areas of Agreement / Disagreement

Participants express differing views on the boundedness of intervals and its implications for proving that set A has measure zero. There is no consensus on the method proposed or its validity.

Contextual Notes

Limitations include the dependence on the properties of open intervals and the assumptions regarding the boundedness of set E. The discussion does not resolve the mathematical steps necessary to prove the measure zero of set A.

arvindam
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A set E subset of real numbers has measure zero. Set A={x2 : x\inE}. How to prove that set A has measure zero?
(E could be any unbounded subset of R)
 
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I would approximate E by open intervals. That is: there exists open intervals ]a_i,b_i[ such that

E\subseteq \bigcup]a_i,b_i[

and such that

\sum{b_i-a_i}<\epsilon.

Now square the open intervals to obtain an approximation of A.
 
But intervals are not bounded. b2i-a2i does not have a bound when b--ai is bounded.
 
arvindam said:
But intervals are not bounded. b2i-a2i does not have a bound when b--ai is bounded.

it is easy for any bounded set of measure zero. Split your set up into countably many of these.
 

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