Jordan-Holder Theorem and Grothendieck Group.

Let $\mathcal{A}$$\mathcal{A}$ be an abelian category such that each object has finite length. For example, the category of finite-dimensional vector spaces.

Then let us consider $K_0(\mathcal{A})$$K_0(\mathcal{A})$. By the Jordan-Hölder theorem, we know that $K_0(\mathcal{A})$$K_0(\mathcal{A})$ is a free abelian group, $K_0(\mathcal{A})\cong \underset{[S]\in \mathrm{Irr}(\mathcal{A})}{⨁}\mathbb{Z}\cdot [S]$$K_0(\mathcal{A})\cong\bigoplus_{[S]\in\mathrm{Irr}(\mathcal{A})}\mathbb{Z}\cdot[S]$.

Then the length $\ell$$\ell$ is an abelian group homomorphism $\ell :K_0(\mathcal{A})\to \mathbb{Z}$$\ell:K_0(\mathcal{A})\to\mathbb{Z}$. If you like analysis, you can extend it as follows:

Consider the set of equivalence classes of irreducible objects in $\mathcal{A}$$\mathcal{A}$, denoted by $\mathrm{Irr}(\mathcal{A})$$\mathrm{Irr}(\mathcal{A})$. Equip $\mathrm{Irr}(\mathcal{A})$$\mathrm{Irr}(\mathcal{A})$ with the counting measure.

Then the length could be extended to $\begin{array}{c}\int :L^1(\mathrm{Irr}(\mathcal{A}))\to \mathbb{C}\end{array}$$\int:L^1(\mathrm{Irr}(\mathcal{A}))\to\mathbb{C}\\$.

The multiplicity $\mu (S,V)$$\mu(S,V)$ is just the projection map or the coordinate function at $[S]$$[S]$.