Description of $\tau $-smooth idempotent measures

In this talk, for a Tychonoff space $\Omega $, we study the space ${{I}_{\tau }}\left( \Omega \right)$ of idempotent probability $\tau $-smooth measures on $\Omega $. In the set of idempotent probability measures, the base of the product topology is introduced and it is shown that for a compact Hausdorff space $\Omega $ the topological space $I\left( \Omega \right)$ is also a compact Hausdorff space. Then we establish that the space $I\left( \Omega \right)$ of idempotent probability measures is Čech-complete if and only if the given Tychonoff space $\Omega $ is Čech-complete. For a compact Hausdorff space $\Omega $, we established a space ${{I}_{\mathfrak{B}}}\left( \Omega \right)$ of all idempotent probability measures on $\Omega $, which define as set-funtions on the $\sigma $-algebra of Borel sets in $\Omega $, and a space ${{I}_{C}}\left( \Omega \right)$ of all normed max-plus linear functionals on the set of all continuous functions on $\Omega $, equipped with idempotent operations. The main result declares that the spaces ${{I}_{\mathfrak{B}}}\left( \Omega \right)$ and ${{I}_{C}}\left( \Omega \right)$ are homeomorphic. Moreover, for a Tychonoff space $\Omega $, we proved max-plus variant of the Riesz representation theorem. In addition to, for a Tychonoff space $\Omega $, we proved max-plus variant of the Riesz representation theorem. From this, the main result of the research work: A description of $\tau $-smooth idempotent measures in Tychonoff space $\Omega $ as a set function is obtained.