The first Brauer-Thrall conjecture for extriangulated length categories

Let $(\mathcal{A},\mathrm{\Theta })$ be a length category. We introduce the notation of Gabriel-Roiter measure with respect to $\mathrm{\Theta }$ and extend Gabriel’s main property to this setting. Using this measure, when $(\mathcal{A},\mathrm{\Theta })$ satisfies some technical conditions, we prove that $\mathcal{A}$ has an infinite number of pairwise nonisomorphic indecomposable objects if and only if it has indecomposable objects of arbitrarily large length. That is, the first Brauer-Thrall conjecture holds.