Abstract. A new type of fractal measures Xs, 1 < s < 2, defined on the subsets of the graph of a continuous function is introduced. The ^-dimension defined by this measure is 'closer' to the Hausdorff dimension than the other fractal dimensions in recent literatures. For the Weierstrass type functions defined by W(x) = £S°'k-aig()Jx), where X > 1 , 0 < a < 1 , and g is an almost periodic Lipschitz function of order greater than a , it is shown that thê -dimension of the graph of W equals to 2 -a , this conclusion is also equivalent to certain rate of the local oscillation of the function. Some problems on the ' knot ' points and the nondifferentiability of W are also discussed.

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