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The main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions , where  is the family of all balls centred on E and α is a real parameter. With mild assumptions on φα, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set V in ℝ D , these notions are used to introduce the interior and exterior measures and dimensions of any Borel subset of ∂V. We stress that these dimensions depend on the choice of φα. Two determining functions are considered, φα(B)=Vol D (B∩V)diam(B)α-D  and φα(B)=Vol D (B∩V)α/D , where Vol D  denotes the D-dimensional volume.

General Hausdorff functions, and the notion of one-sided measure and dimension