Multiplicities of weakly graded families of ideals

In this article, we extend the notion of multiplicity for weakly graded families of ideals which are bounded below linearly. In particular, we show that the limit $e_W(\mathfrak{I}):=\lim\limits_{n\to\infty}d!\frac{\ell_R(R/I_n)}{n^d}$ exists where $\mathfrak I=\{I_n\}$ is a bounded below linearly weakly graded families of ideals in a Noetherian local ring $(R,\mathfrak m)$ of dimension $d\geq 1$ with $\dim(N(\hat{R}))<d$. Furthermore, we prove that ``volume=multiplicity" formula and Minkowski inequality hold for such families of ideals. We explore some properties of $e_W(\mathfrak J)$ for weakly graded families of ideals of the form $\mathfrak J=\{(I_n:K)\}$ where $\{I_n\}$ is an $\mathfrak m$-primary graded family of ideals. We provide a necessary and sufficient condition for the equality in Minkowski inequality for the weakly graded family of ideals of the form $\mathfrak J=\{(I_n:K)\}$ where $\{I_n\}$ is a bounded filtration. Moreover, we generalize a result of Rees characterizing the inclusion of ideals with the same multiplicities for the above families of ideals. Finally, we investigate the asymptotic behaviour of the length function $\ell_R(H_{\mathfrak m}^0(R/(I_n:K)))$ where $\{I_n\}$ is a filtration of ideals (not necessarily $\mathfrak m$-primary).