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Let w = w(z) be any K-quasiconformai mapping (in the sense of Pfluger-Ahlfors) of a domain D in the z-plane into the w-plane. Since w = w(z) is a measurable mapping (vid. Bers [1]), it transforms any set of Hausdorffs 2-dimensional measure zero in D into such another one. However, A. Mori [5] showed that for 0 < α ≤ 2, any set of -dimensional measure zero in a Jordan domain D is transformed by w = w(z)into a set of α-dimensional measure zero.