I was following along a Y-T video proving that there is no useful measure on the power set of R . The proof is too long to lay out here but I'm certain that most respondents here will recognize it if I just describe it.The proof starts by pulling all the rational numbers from (0,1], creating little `boxes' along the interval. Next construct a set A that contains one and only one point from each box. Then create a sequence, A n by adding a rational r from (-1,1), jogging A left and right in (-1,2]. Then we take the countable union of the A n , and here's where I lose the plot. The claim, setting up the final blow in the proof, is that(0,1] ⊆ { ∪ A n , n ∈ N } ⊆ ( − 1 , 2 ]I get the RHS but the LHS looks exactly wrong to me. (0,1] contains all the reals and by construction the term in the middle is devoid of the rational numbers. How can the term in the middle contain (0,1]?

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I was following along a Y-T video proving that there is no useful measure on the power set of       R  . The proof is too long to lay out here but I&#039;m certain that most respondents here will recognize it if I just describe it.The proof starts by pulling all the rational numbers from (0,1], creating little `boxes&#039; along the interval. Next construct a set A that contains one and only one point from each box. Then create a sequence,       A    n   by adding a rational r from (-1,1), jogging A left and right in (-1,2]. Then we take the countable union of the       A    n  , and here&#039;s where I lose the plot. The claim, setting up the final blow in the proof, is that(0,1]   ⊆  {  ∪      A    n    ,  n  ∈      N    }  ⊆  (  −  1  ,  2  ]I get the RHS but the LHS looks exactly wrong to me. (0,1] contains all the reals and by construction the term in the middle is devoid of the rational numbers. How can the term in the middle contain (0,1]?

Haleigh Vega · Accepted Answer

My claim about the middle term is wrong. While it&#039;s true that the set A is constructed to exclude the rational numbers in (0,1], the       A    n   are constructed by adding the rational numbers in (-1,1] to A, thus restoring those removed.This closes the question unless someone has something else to add.

I was following along a Y-T video proving that there is no useful measure on the power set of <m

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