I am a graduate student of Mathematics.In our measure theory course I encountered the definition of Lebesgue measurable set which is as follows:(1) A set A ⊂ R is Lebesgue measurable if ∃a closed set B ⊂ A such that | A − B | = 0.I want to know whether the following is equivalent to the above definition:(2) A set A is Lebesgue measurable if for each ϵ &gt; 0 there exists a closed set B ϵ ⊂ A such that | A − B ϵ | &lt; ϵ.Clearly (1) implies (2) but is the other direction true?

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I am a graduate student of Mathematics.In our measure theory course I encountered the definition of Lebesgue measurable set which is as follows:(1) A set   A  ⊂      R   is Lebesgue measurable if   ∃a closed set   B  ⊂  A such that       |    A  −  B      |    =  0.I want to know whether the following is equivalent to the above definition:(2) A set   A is Lebesgue measurable if for each   ϵ  &amp;gt;  0 there exists a closed set       B    ϵ    ⊂  A such that       |    A  −      B    ϵ        |    &amp;lt;  ϵ.Clearly (1) implies (2) but is the other direction true?

Elias Flores · Accepted Answer

The first definition is (strange and ) not equivalent to the second one (which agrees with the usual definition). Let   A  =      R    ∖  {  0  }. Certainly   A is Lebesgue measurable according any standard definition and it does satisfy the property in (2). But there is no closed set   B  ⊂  A with       |    A  −  B      |    =  0. This is because       |        B          c            |    =  0 (:       B          c        ⊂  (  A  −  B  )  ∪  {  0  }) which implies that   B is dense. But   B is closed, so   B  =      R  . This contradicts the fact that   B  ⊂  A.

I am a graduate student of Mathematics.In our measure theory course I encountered the definition of

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