On one-sided densities of arcs of positive two-dimensional measure

Extract

1. Let AB be a plane simple arc, which, considered as a point-set, has a positive two-dimensional measure. If x be a point of AB, a point x′ ≠ x is said to be to the left or to the right from x according as it belongs to the arc Ax or xB. The upper and the lower limits of the ratio [m₂{Ax C(x, r)}]/πr², where C(x, r) denotes the closed disc with centre x and radius r, as r → 0 define the upper and the lower left densities at the point x. When the two densities are equal their common value defines the left density. The right densities are defined similarly.