00 An = nnAn E M .

8 Comparing Borel Sets 19 so that specializing to the Borel sets on the real line we get B(O, 1] = B(lR) n (0, 1]. Proof. (1) We proceed in a series of steps to verify the postulates defining a afield. (i) First, observe that no e Bo since nno = no and n E B. (ii) Next, we have that if B = A no e Bo, then no\ B =no\ A no= no(n \A) e Bo since n \A e B. (iii) Finally, if for n :::: 1 we have Bn = An no, and An e B, then u =u n=l Bn n=l (U An) n no e Bo ()() ()() ()() Anno= n=l since Un An e B. (2) Now we show a(Co) = a(C) n no.

N-+oo (ii) If An -+ A, then P(An) -+ P(A). Proof of 8. (ii) follows from (i) since, if An -+ A, then limsupAn = liminfAn =A. n-+oo n-+oo Suppose (i) is true. Then we get P(A) = P(lim inf An) n-+oo ~ limsupP(An) n-+oo ~ ~ lim inf P(An) n-+oo P(limsupAn) n-+oo = P(A), so equality pertains throughout. _n (from continuity property 7) :::: lim sup P(An). n-+00 completing the proof. 1 Let Q D = IR, and suppose P is a probability measure on JR. Define F(x) by F(x) = P((-oo,x]), x e JR. 3) Then (i) F is right continuous, (ii) F is monotone non-decreasing, (iii) F has limits at ±oo F(oo) := lim F(x) = 1 xtoo F(-oo) := lim F(x) = 0.

### A Probability Path by Sidney I. Resnick

by David

4.3