##
**On the structure of finite level and \(\omega\)-decomposable.**
*(English)*
Zbl 1323.03065

J. E. Jayne and C. A. Rogers [J. Math. Pures Appl. (9) 61, 177–205 (1982; Zbl 0514.54026)] proved that every function from an analytic space to a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each \(F_\sigma\) set is again \(F_\sigma\). The main motivation of this paper is to study some generalizations of the Jayne-Rogers theorem.

In this paper, \(\Sigma_{\alpha,\beta}\) functions are defined as functions \(f\) with \(f^{-1}(S)\in\Sigma^0_\alpha\) for every \(S\in\Sigma^0_\beta\). The author first studies some properties of \(\Sigma_{\alpha,\beta}\) functions. Then, the author uses this notion to investigate the possibility for decomposing a function into countably many \(\Sigma^0_\beta\)-measurable functions. The author mostly focuses on finite level functions and the possibility for decomposing a \(\Sigma_{n,n}\) function into countably many continuous functions with \(\Delta^0_n\) domains. Some interesting equivalent conditions of generalizations of the Jayne-Rogers theorem are given. Solecki’s dichotomy theorem on \(\omega\)-decomposable \(\Sigma^0_2\)-measurable functions is generalized onto finite level Borel functions.

The author also presents several relevant questions in the paper.

In this paper, \(\Sigma_{\alpha,\beta}\) functions are defined as functions \(f\) with \(f^{-1}(S)\in\Sigma^0_\alpha\) for every \(S\in\Sigma^0_\beta\). The author first studies some properties of \(\Sigma_{\alpha,\beta}\) functions. Then, the author uses this notion to investigate the possibility for decomposing a function into countably many \(\Sigma^0_\beta\)-measurable functions. The author mostly focuses on finite level functions and the possibility for decomposing a \(\Sigma_{n,n}\) function into countably many continuous functions with \(\Delta^0_n\) domains. Some interesting equivalent conditions of generalizations of the Jayne-Rogers theorem are given. Solecki’s dichotomy theorem on \(\omega\)-decomposable \(\Sigma^0_2\)-measurable functions is generalized onto finite level Borel functions.

The author also presents several relevant questions in the paper.

Reviewer: Longyun Ding (Tianjin)

### MSC:

03E15 | Descriptive set theory |

54H05 | Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) |

26A21 | Classification of real functions; Baire classification of sets and functions |