Let S be a foundation topological semigroup and Ma(S)\documentclass[12pt]{minimal}
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\begin{document}$$M_a(S)$$\end{document} the space of all measures μ∈M(S)\documentclass[12pt]{minimal}
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\begin{document}$$\mu \in M(S)$$\end{document} for which the maps x⟼|μ|∗δx\documentclass[12pt]{minimal}
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\begin{document}$$x\longmapsto |\mu |*\delta _{x}$$\end{document} and x⟼δx∗|μ|\documentclass[12pt]{minimal}
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\begin{document}$$x\longmapsto \delta _{x}*|\mu |$$\end{document} from S into M(S) are weakly continuous. In the present paper, we introduce and study the concept of ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi$$\end{document}-amenability for S and investigate the relations between ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi$$\end{document}-amenability of S and essential ϕ^\documentclass[12pt]{minimal}
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\begin{document}$$\widehat{\phi }$$\end{document}-amenability of Ma(S)\documentclass[12pt]{minimal}
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\begin{document}$$M_a(S)$$\end{document}, where ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi$$\end{document} is a character on S and ϕ^\documentclass[12pt]{minimal}
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\begin{document}$$\widehat{\phi }$$\end{document} is the extension of ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi$$\end{document} to Ma(S)\documentclass[12pt]{minimal}
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\begin{document}$$M_a(S)$$\end{document}.
