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Extra resources for Uniform Scalar Integrability and Strong Laws of Large Numbers for Pettis Integrable Functions with Values in a Separable Locally Convex Space
We denote by B E$ the closed unit ball of E$=E. The uniform structure on MU 1E =MW 1E =MP 1E generated by the semi-distances d A (A # S) is also generated by the only distance n d=d BE $ . Let P n =2 &n 2k=1 $ ek (n # N*). The set D(P n , $ 0 ) has a single element P n $ 0 and we have d(P n , $ 0 )= sup x$ # BE $ = sup x$ # BE $ || |( x$, x& y) | dP (x) d$ ( y) n 0 | |( x$, x) | dP (x) n 2n = sup 2 x$ # BE $ &n : |( x$, e k ) | k=1 2 &nÂ2 S-Uniform Scalar Integrability and SLLN 123 thus lim n Ä d(P n , $ 0 )=0.
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