What does it mean that a function in Sobolev spaces (for example) to be defined only upto a set of measure zero?

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Misael Li · Accepted Answer

There&#039;s nothing particular about Sobolev spaces here. If   (  X  ,      A    ,  μ  ) is a measure space and   Y is any set, we can define an equivalence relation on the set       F    (  X  ,  Y  ) of all functions from   X to   Y by saying that   f  ∼  g if  μ            (        {  x  ∈  X  ∣  f  (  x  )  ≠  g  (  x  )  }            )        =  0.Check that this is an equivalence relation. The very common abuse of notation is to denote the equivalence class [  f] of   f by   f itself. Saying that   f is defined up to a set of measure zero means, effectively, specifying the equivalence class but not actually choosing a representative for such class.

What does it mean that a function in Sobolev spaces

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