  We study various measure theories using the classical approach and then
compute the Hausdorff dimension of some simple objects and self-similar
fractals. We then develop a nonstandard approach to these measure theories and
examine the Hausdorff measure in more detail. We choose to study Hausdorff
measure over any other measures since it is well-defined for all sets, and
widely used in many different areas in mathematics, physics, probability
theory, and so forth.
  Finally we generate a particular set and compute its upper and lower
Hausdorff dimension. We compare our set with box-counting dimension and other
well-known fractal behaviors to analyze the set in a greater detail.
