In this paper, we prove that if a commuting family of operators \( \tau = (T_\lambda)_{\lambda\in\Lambda}\) on a Banach space \(\mathcal{X}\) is Bochner integrable, then \(r\left({}^{^{B}}\!\!\!\!\int_{\Lambda}T_{\lambda}d\mu(\lambda) \right) \leq \int_\Lambda r(T_\lambda)\,d\mu(\lambda).\) This result extends the well-known theorem on the subadditivity of the spectral radius for a finite set of commuting operators. We also provide an example illustrating that Bochner integrability cannot be substituted by a weaker form of integrability.
