  The objective of this paper is to study the Hausdorff measure, a fundamental
concept in fractal geometry. We begin by defining the Hausdorff outer measure,
which allows us to assign a measure to any subset of a metric space. We then
present Caratheodory's theorem, which characterizes the outer measure in terms
of coverings by open sets. We also study the construction of the Hausdorff
measure from the outer measure. Next, we introduce the Hausdorff measure
proper, which is an extension of the outer measure. We give a formal definition
and discuss its properties, such as monotonicity and translation invariance. We
also address the notion of Hausdorff dimension, which is a generalization of
Euclidean dimension to non-regular sets. Finally, we present an application of
the Hausdorff measure to the Cantor ternary set, which is a classic example of
a fractal set. We calculate the Hausdorff dimension of this set and show how
the Hausdorff measure can be used to quantify the geometric complexity of this
object. Overall, this paper provides a comprehensive introduction to the
Hausdorff measure, which is an essential concept for understanding fractal
geometry and its applications in many scientific fields.
