The nonstandard characterization of uniformly integrable functions;

For weighted sums of the form k nj=1a nj d jX,where{a nj ,1≤j≤k n↑∞} is a real constants array and {d nX,n≥1} is martingale difference series,we establish the relationship between the convergence and the p\|smoothable Banach space under the condition of {a nj }\|uniform integrability,andwe get the strong law of large numbers for weighted sums of martigale difference series.

On this basis the necessary and sufficient conditions to the uniform integrability of sequences of fuzzy valued functions were given,and the implication relations between uniform integrability of sequences of fuzzy valued functions and the uniformly boundedness of the their fuzzy valued integrals were studied.

the paper simplifies the proofs of theorems in referenc e and further studies the conditions of uniform differentiability of a function.

The uniform differentiability of characteristic function and conditional characteristic function and reversion formula of conditional eigenfunction are obtained in this paper.