University of Qom Measure Algebras and Applications 3116-014X 3 1 2025 06 01 Matrix-valued measures and integration on foundation semigroups 18 31 3999 10.22091/maa.2025.14717.1042 EN Ali Jabbari Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan 0000-0003-4273-1998 Journal Article 2025 11 29 This paper develops a comprehensive theory of matrix-valued measures and integration on foundation topological semigroups, extending the well-established framework for topological groups. We establish fundamental results, including polar decomposition, duality theory, and convolution algebras for $M_n$-valued measures. The paper characterizes several important classes of measures ($\Mel$, $\Mer$, $\Mal$, $\Mar$) and analyzes their structural properties, showing that $\Ma$ forms an L-ideal and $\Me$ is an L-subalgebra. Finally, we develop a theory of quasi-invariant matrix-valued measures, proving existence results and characterizing the measure algebra $\Me$ in terms of equi-quasi-invariant measures. Our work provides a unified framework for non-commutative harmonic analysis on semigroups, generalizing both the scalar theory for foundation semigroups and the matrix-valued theory for groups. https://maa.qom.ac.ir/article_3999_8e62dbccaaa7913121a7af9527a720a4.pdf