Measurement properties of random fractals

This dissertation is concerned with the Hausdorff and packing measures of some random sets, and some properties of Borel measures in R^d.

Chapter One contains essential definitions, notations and known results which will be used later.

Chapter Two deals with Kaufman's dimension and packing measure of product sets. In the end some measure properties of projections are discussed.

Chapter Three is concerned with dimensions of finite Borel measures in R^d and properties of product measures and projections.

In Chapter Four we study the Hausdorff and packing measures of the product of the two zero sets of independent stable processes with indices between 1 and 2, and also the Hausdorff measure of the projection.

Chapter Five is involved with the Hausdorff and packing measures of the random Cantor set. In the end, we generalize the result to the random closed sets belonging to regular sequences. Finally the packing dimension of general random closed sets belonging to decreasing convergent sequences has been found.