In this paper, we show that the lower dimension is not invariant under quasi-Lipschitz mapping, and then we find an invariant named the quasi-lower dimension. We also compute the quasi-lower dimension of a class of sets defined by digit restrictions, and then give an example to distinguish the quasi-lower dimension and other dimensions.

Given metric spaces E and F , it is well known thatwhere dimH E, dimP E, dim B E, dimBE denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of E, respectively. In this note we shall provide examples of compact sets showing that the dimension of the product E ×F may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of the product of sets defined by digit restrictions.

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We obtain a complete description for a probability measure to be doubling on an arbitrarily given uniform Cantor set. The question of which doubling measures on such a Cantor set can be extended to a doubling measure on [0; 1] is also considered

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