The generalized continuous model theory, Borel complexity and stability

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