Summary: Math­571. 4­th Homework. Due Tuesday, March 20, 2007.
1. Let L0(, F, P) be the vector space consisting by the r.v.'s defined on the probability
space (, F, P). Let L0(, F, P) be the classes of equivalence of L0(, F, P) with
respect to the equivalence relation X Y iff X = Y a.s. Show that there is no
norm · on L0(, F, P) such that Xn
P
X iff Xn - X 0.
2. Prove that there exists no distance d in L0(, F, µ) such that d(Xn, X) 0 iff
Xn
a.s.
X.
3. Consider the probability space ([0, 1], B([0, 1]), m), where m is the Lebesgue measure.
Let Xn = n
log n
I(0, 1
n
), n 3. Show that {Xn}
n=1 is uniformly integrable, Xn
Pr
0

Source: Arcones, Miguel A. - Department of Mathematical Sciences, State University of New York at Binghamton

Collections: Mathematics