Measure Theory on Hausdorff Measures

Let y = h(x) be defined for 0 < x < oo and assume values in 0 ^ y ^ + co. Let S be any linear set of points and p an arbitrary positive number. Cover S by a countable number of open intervals h , I2, • • • of lengths xx, x2 , • • • each of which is less than p, and denote by mp(S; h) the lower bound of h(xi) + h(x2) + • • • for all such coverings of S. Then m(S; h) = limp|0mp(/S; h) is called the (linear, exterior) Hausdorff measure of S with respect to h(x). (F. Hausdorff [Math. Ann. vol. 79 (1918) pp. 157-179] originally considered the most important case of continuous monotone h(x) with \imxioh(x) = 0. For later studies see e.g. G. Bouligand [Les definitions modernes de la dimension, Actualités scientifiques et industrielles, no. 274, Paris, 1935] and A. Dvoretzky IProc. Cambridge Philos. Soc. vol. 44 (1948) pp. 13-16].) If m(S; h) < co implies ra(#; g) < oo we write g < h. If either g < h or h < g we say that g ix) and h(x) are comparable; if both relations hold we write g ~ h (read: equivalent). If miS; g) = m(S; h) for all S we write g œ h (read: strictly equivalent). Given any hix) it can be shown that there exists a continuous monotone (nondecreasing) g(x) strictly equivalent to hix). Whatever hix) we put h*ix) = x inf0<^aJ h(t)/t. Then h* ~ h or, more precisely, for all S we have m(S] h*) ^ m(S', h) ^ 2m(S\ h*). Using this result it can be shown that g < h if and only if