By Robin Hartshorne, C. Musili

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**Example text**

Let E . Then a(& x B2) = ( B , x 3232) u ( d B 1 x B,) . B1 Therefore we have the following three cases to consider. (1) B1 is a p-continuity set and B2 is a v-continuity set. (3) v(B,) = 0. 7 we have x B2 E Convolution of measures 33 and similarly in the remaining case. Thus X(B1 x B2) = ( p 8 u)(B1 x B2) for all B1 x B2 E E and hence X = p 8 u. Hence the rw-relatively compact sequence ( p n 8 un)nyl has a unique cluster point p 8 U, and it follows that rn rW- limn p n 8 vn = /L 8 V. 9 Let ( E , d ) be a separable complete metric Abelian , *) with the convolution * defined group.

Since each K N is compact, n { K N : N E A@)} # 0. But for y E n { K N : N E N ( E ) } we have p ~ ( y = ) p ~ ( x whenever ) N E N ( E ) . From n { N : N E N ( E ) }= { 0 } we have x = y E K and consequently K1 c K . 2 Let p E M b ( E ) . i(a) := e+'")p(dx) -+ C given /' for all a E E' is called the Fourier transform of p. For Banach spaces E , F denote by L ( E ,F ) the set of all continuous linear mappings from E into F , and consider T E L ( E , F ) . The adjoint T t of T is continuous linear mapping from F' into E' given by (2,Ttb) = ( T z ,b) whenever x E E , b E F'.

M p(B,) = 0. 2 Let satisfying (pn)nll be a n increasing sequence in M b ( E ) suppn(E) < nrl T h e n sup,>1 - pn E Adb@). - The Prohorov theorem 23 Proof. Write p ( B ) := supn,,pn(B) for all B E 23(E). 1. 3 Lei! ( E , d ) be a compact metric space. T h e n for each a>O M q E ) := { p E M b ( E ): p ( E ) 5 a) is rw -compact. Proof. Hence is weakly compact. 3 the mapping is a bijection from M ( a ) ( E )onto V p ) . r,-compact. 4 A set H (b) to each E c M b ( E ) is called uniformly tight if > 0 there exists a compact set K c E such that p ( E \ K ) < & forall p u H .