matrix-valued measures and integration on foundation semigroups

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 (mle (s, mn), mre (s, mn), mla (s, mn), mra (s, mn)) and analyzes their structural properties, showing that ma(s, mn) forms an l-ideal and me(s, mn) is an l-subalgebra. finally, we develop a theory of quasi-invariant matrix-valued measures, proving existence results and characterizing the measure algebra me(s, mn) 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.