Generalizations Of Some Classical Properties In Framework Of G-expectation

In classicall measure theory,Egoroff's theorem,Lusin's theorem and continuous function approximation theorem are three of the important,theorems.The ?-additivity of measure is used to prove these theorems,which plays a crucial role in the proof.The ?-additivity is not,satisfied in the framework of capacity which causes great difficulties to study Egoroff's theorem and Lusin's theorem.Many researchers have done lots of studies on Egoroffs theorem and Lusin's theorem when the measure is monotone and non-additive.see Li(2003).Li and Yasuda(2004,2005),Li and Mesiar(2011).This thesis focuses on studying Egoroff's theorem and Lusin's theorem under the meaning of the g-expectation capacity.The g-expectation is nonlinear expectation in Peng(1997),and can be viewed as a nonlinear extension of the well-known Girsanov transformation,which is very important for expected utility theory in economy.The g-expectation is widely applied in financial markets which are incomplete and irregular in mathematical finance,see El Karoui(1997).Supposing the generator g satisfies the conditions of Lipschitz.for any y,g(t.y.0)? 0,and sub-additivity about y,z,this thesis proves the capacity Vg satisfying continuity and sub-?-additivity.which is used to prove Egoroff's theorem.Under ther above-mentioned conditions of g.given a functional space of Brownian motion,this thesis proves Lusin's theorcem by sub-?-additivity and Egoroff's theorem.