Borel* Sets in the Generalised Baire Space

We start by giving a survey to the theory of Borel*(κ) sets in the generalized Baire space Baire(κ) = κ^κ. In particular we look at the relation of this complexity class to other complexity classes which we denote by Borel(κ), Δ^1_1(κ) and Σ^1_1(κ) and the connections between Borel*(κ)-sets and the infinitely deep language M_{κ^+κ}. In the end of the paper we prove the consistency of Borel*(κ) \ne Σ^1_1(κ).