The generalized continuous model theory, Borel complexity and stability

Given Polish space $\mathcal{Y}$ and a continuous language $L$ we study the corresponding logic $\mathsf{I}\mathsf{s}\mathsf{o}(\mathcal{Y})$-space ${\mathcal{Y}}_L$. We build a framework of generalized model theory towards analysis of Borel complexity of families of subsets of Effros spaces $\mathcal{F}(\mathcal{Y}{)}_L^k\times \mathcal{F}(\mathsf{I}\mathsf{s}\mathsf{o}(\mathcal{Y}){)}^l$ corresponding to standard model-theoretic properties. In this paper we mainly apply this approach to stability.