Suppose E ⊂ R d and O n is the open set O n = { x ∈ R d : d ( x , E ) &lt; 1 n } If E is compact, then m ( E ) = lim n → ∞ m ( O n ) . I wonder what are some examples that this equality does not hold when E is closed and unbounded, or E is open and bounded.

Question

Suppose   E  ⊂            R              d       and             O              n       is the open set                                          O                                n                          =                  {          x          ∈                                    R                                      d                                :          d          (          x          ,          E          )          &amp;lt;                      1            n                    }                    If   E is compact, then   m  (  E  )  =      lim          n      →      ∞        m      (                  O                    n              )  . I wonder what are some examples that this equality does not hold when   E is closed and unbounded, or   E is open and bounded.

Charlee Gentry · Accepted Answer

For the case of open and bounded   E take   E  =  [  0  ,  1  ]  ∖  C where   C is a fat Cantor set with   0  &amp;lt;  m  (  C  )  &amp;lt;  1. Then   m  (      O    n    )  →  m  (      E    ¯    )  =  1 whereas   m  (  E  )  &amp;lt;  1.

rjawbreakerca · Accepted Answer

For closed and unbounded case:Take   d  =  1,   E  =      N  . Then       O    n    =      ⋃          k      ∈              N              (  k  −  1      /    n  ,  k  +  1      /    n  ), where the sets are pairwise disjoint for   n  ≥  2, for example.For any fixed   n  ≥  2,   m  (      O    n    )  =      ∑          k      =      1        ∞        2    n    =      2    n        ∑          k      =      1        ∞    1  =  +  ∞. But   m  (      N    )  =  0.PS: In the general case, take   E to be any sequence of different points in             R        d   going to infinity.

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