  A density function for an algebraic invariant is a measurable function on
$\mathbb{R}$ which measures the invariant on an $\mathbb{R}$-scale. This
function carries a lot more information related to the invariant without
seeking extra data. It has turned out to be a useful tool, which was introduced
by the third author, to study the characteristic $p$ invariant, namely
Hilbert-Kunz multiplicity of a graded ${\bf m}$-primary ideal.
  Here we construct density functions $f_{A,\{I_n\}}$ for a Noetherian
filtration $\{I_n\}_{n\in\mathbb{N}}$ of graded ideals and
$f_{A,\{\widetilde{I^n}\}}$ for a filtration given by the saturation of powers
of a graded ideal $I$ in a standard graded domain $A$. As a consequence, we get
a density function $f_{\varepsilon(I)}$ for the epsilon multiplicity
$\varepsilon(I)$ of a graded ideal $I$ in $A$. We further show that the
function $f_{A,\{I_n\}}$ is continuous everywhere except possibly at one point,
and $f_{A,\{\widetilde{I^n}\}}$ is a continuous function everywhere and is
continuously differentiable except possibly at one point. As a corollary the
epsilon density function $f_{\varepsilon(I)}$ is a compactly supported
continuous function on $\mathbb{R}$ except at one point, such that
$\int_{\mathbb{R}_{\geq 0}} f_{\varepsilon(I)} = \varepsilon(I)$.
  All the three functions $f_{A,\{I^n\}}$, $f_{A,\{\widetilde{I^n}\}}$ and
$f_{\varepsilon(I)}$ remain invariant under passage to the integral closure of
$I$.
  As a corollary of this theory, we express the mixed multiplicities
$e_i(A[It])$ of the Rees algebra of a graded ideal $I$ in terms of the
intersection numbers of two Cartier divisors on $X=\mathbf{Proj}
\left(\oplus_{n\geq 0}\mathcal{I}^n\right)$, where $\mathcal{I}$ is the ideal
sheaf associated to $I$ on $\mathrm{Proj}~A$.
