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Let
 be a measurable set. We define the essential boundary of  as
where  is the Lebesgue measure.
Compare the definition of
 with the definition of the topological boundary
 which can be written as
Hence one clearly has
 .
Notice that the essential boundary does not depend on the Lebesgue representative of the set  , in the sense that if
 then
 . For example if
 is the set of points with rational coordinates, one has
 while
 .
Nevertheless one can easily prove that
 is always a closed set (in the usual sense).
Moreover one has
 (where
 is the reduced boundary and
 is the  -dimensional Hausdorff measure) and hence a unit normal vector is defined in
 -a.e.
 .
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