9.1 Introduction to ``hyperfinite'' DST

    9.1a General set-up 
    9.1b Comments on notation 
    9.1c Borel and projective sets in a nonstandard domain
    9.1d Some applications of countable Saturation     
    9.1e Operation A and Souslin sets
    
9.2 Operations, countably determined sets, shadows

    9.2a Operations and quantifiers 
    9.2b Countably determined sets 
    9.2c Shadows or standard part maps
    
9.3 Structure of the hierarchies

    9.3a Operations associated with Borel and projective classes 
    9.3b The ``shadow'' theorem 
    9.3c Closure properties of the classes
    
9.4 Some classical questions

    9.4a Separation and reduction 
    9.4b Countably determined sets with countable cross-sections 
    9.4c Countably determined sets with internal and \Sigma^0_1 cross-sections 
    9.4d Uniformization 
    9.4e Variations on Louveau's theme 
    9.4f On sets with bf \Pi^0_1 cross-sections 
    
9.5 Loeb measures

    9.5a Definitions and examples 
    9.5b Loeb measurability of projective sets 
    9.5c Approximations almost everywhere 
    9.5d Randomness in a hyperfinite domain 
    9.5e Law of Large Numbers 
    9.5f Random sequences and hyperfinite gambling
    
9.6 Borel and countably determined cardinalities

    9.6a Preliminaries 
    9.6b Borel cardinals and cuts 
    9.6c Proof of the theorem on Borel cardinalities 
    9.6d Complete classification of Borel cardinalities  
    9.6e Countably determined cardinalities
    
9.7 Equivalence relations and quotients

    9.7a Silver's theorem for countably determined relations 
    9.7b Application: nonstandard partition calculus                                  
    9.7c Generalization
    9.7d Transversals of ``countable'' equivalence relations 
    9.7e Equivalence relations of monad partitions 
    9.7f Borel and countably determined reducibility 
    9.7g Reducibility structure of monad partitions 

Historical and other notes to Chapter 9