Multiplicities and density functions in commutative algebra

In this expository article we discuss the numerical invariant Hilbert-Kunz multiplicity and provide numerical characterizations of integral dependence of ideals, where the main focus is on the graded set up. To emphasize the subtlety of the characteristic p techniques to non-expert readers, we give some self contained proofs and sketches of some proofs. In the later part we give an introduction of the theory of density functions and their applications to Hilbert-Kunz multiplicities and briefly describe how these functions along with basic analysis provide numerical characterizations of integral dependence of ideals.