Let $$(R, {\mathfrak {m}}, k)$$ be a complete CohenâMacaulay local ring. In this paper, we assign a numerical invariant, for any balanced big CohenâMacaulay module, called $$\underline{\mathsf {h}}$$ -length. Among other results, it is proved that, for a given balanced big CohenâMacaulay R-module M with an $${\mathfrak {m}}$$ -primary cohomological annihilator, if there is a bound on the $$\underline{\mathsf {h}}$$ -length of all modules appearing in $$\mathsf {CM}$$ -support of M, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first BrauerâThrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal CohenâMacaulay modules, our formalism establishes the validity of the conjecture for complete CohenâMacaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big CohenâMacaulay modules is settled. Namely, it is shown that R is of finite $$\mathsf {CM}$$ -type if and only if R is an isolated singularity and the category of all fully decomposable balanced big CohenâMacaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of CohenâMacaulay artin algebras admitting a dualizing bimodule $$\omega $$ , as defined by Auslander and Reiten. It will turn out that, $$\omega $$ -Gorenstein projective modules with bounded $$\mathsf {CM}$$ -support are fully decomposable. In particular, a CohenâMacaulay algebra $$\Lambda $$ is of finite $$\mathsf {CM}$$ -type if and only if every $$\omega $$ -Gorenstein projective module is of finite $$\mathsf {CM}$$ -type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is GabrielâRoiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the GabrielâRoiter (co)measure in the category of maximal CohenâMacaulay modules.