In this paper, we introduce and study a broad class of semi-discrete sampling operators, namely Durrmeyer-type sampling operators \( M^{\eta ,\uptau }_{w,\nu } \) , constructed with respect to a non-negative, locally finite, and non-trivial Borel measure \( \nu \) on \( \mathbb {R} \) , together with a kernel \( \eta \) and an auxiliary weight function \( \uptau \) . This formulation provides a unified framework encompassing weighted and non-uniform sampling scheme. For every locally finite measure \( \nu \) , we establish the fundamental approximation results such as point-wise and uniform convergence on compact subsets of \(\mathbb {R}\) for bounded and uniformly continuous functions. Under an additional admissibility assumption on \( \nu \) , namely quasi-invariance up to a constant multiple under affine transformations, bounded Radon–Nikodým derivative, or absolute continuity with bounded density, we establish the \(L^p(\nu )\) -boundedness and convergence in the \(L^p(\nu )\) -norm for \(1\le p\le \infty \) . We also provide explicit conditions ensuring self-adjointness of operators \(M^{\eta ,\uptau }_{w,\nu }\) in \(L^2(\nu ).\)
