IR are random variables . 2! Let B be the Borel J-algebra of IR. Define U: B- R Such that "H (AE B ) = 1 if ( 0.E ) CA for some 8 7 0 ii ) and u (A ) = O otherwise . ( a) show that 1 is not finitely additive ( b) Let B. = $ ( a,, b, JAU ( Q2 , b2 ] ... U(am, bm] : for real a;'s and bis , and for meIN} Show that the same conditions i) and it ) define u : B. > R which is finitely additive but not countably additive . 3. We have learned that the measure M : J - IR is defined uniquely if u is defined on P, a TT-system Such that F = J (P ) . Show that, if P is not a TT-system but F = (P ), M may not be unique. ( Hint : Enough to consider (2 = [1,2, 3,43 ) Probability 4 . Consider infinite number of cain throws , Le. measure 1 = 1 (a, aza, .... ) | a;E f Head, Tail}} . Let Xp (W) = (number of heads in the first n throws ) = # ( { aj | a; = Head, isn't ) . Find a sequence w = ( a,a, .) e 2 such that lim Xn(w) does not exist . 1 of 1 Dashboard Calendar To Do Notifications Inbox