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 homogeneous ${ \bf m}$-primary ideal. Here we construct density functions $f_{ A,\{ I_n\}}$ for a Noetherian filtration $\{ I_n\}_{ n\in\mathbb{ N}}$ of homogeneous ideals and $f_{ A,\{ \widetilde{ I^n}\}}$ for a filtration given by the saturated powers of a homogeneous 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 homogeneous 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 observe that the `rescaled' Hilbert-Samuel multiplicities of the diagonal subalgebras form a continuous family.
Categories: math.AC math.AG