Suppose ( Ω , F , μ ) is a measure space and f , g are ( F , B ) −−measurable functions where B is the Borel algebra on R . If μ ( { f &lt; g } ) &gt; 0then are we able to find a constant ξ ∈ R such that the set { f ≤ ξ &lt; g } is also of nonzero measure? I am confident that I could find ϵ &gt; 0 such that the set { f &lt; g − ϵ } is of nonzero measure, but I find the existence of such a constant ξ questionable.

Question

Suppose   (  Ω  ,      F    ,  μ  ) is a measure space and   f  ,  g are   (      F    ,      B    )  −−measurable functions where       B   is the Borel algebra on       R  . If  μ  (  {  f  &amp;lt;  g  }  )  &amp;gt;  0then are we able to find a constant   ξ  ∈      R   such that the set   {  f  ≤  ξ  &amp;lt;  g  } is also of nonzero measure? I am confident that I could find   ϵ  &amp;gt;  0 such that the set   {  f  &amp;lt;  g  −  ϵ  } is of nonzero measure, but I find the existence of such a constant   ξ questionable.

delalbaef · Accepted Answer

The rational numbers can be denoted as       r    1    ,      r    2    ,  .  .  .Denote       A    k    =  {  f  ≤      r    k    &amp;lt;  g  }  .Then       ∪    k        A    k    =  {  f  &amp;lt;  g  }.So you can find one       A    k   with positive measure.

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