REDUCTION TO AN AXIOMATIC SYSTEM
EXAMPLES FOR STUDY & PRACTICE
1. a) Prove in general that for any R-valued X with EX2 < ∞, it follows that (EX)2 ≤ EX2.
b) For any R-valued X with (EX)2 < EX2 < 1, prove that there are unique scalars |a| < 1 and 0 that X = a+bZ w. EZ = 0, EZ2 = 1.
c) Given properties i), ii) and iii) exactly as presented in defn.0.0.1 on p.4, verify that it is exactly equivalent to replace iv) as therein stated by
iv)' continuous: Zn ≥ 0, n = 1, 2,...
2. some technical loose ends
a) For S = as in Lemma 0.0.2, p.11, verify that
b) Verify Eqn.(11) p.16, that it is actually true that any non-negative random variable Z may be expressed as the supremum of all the non-negative simple functions below it:
c) Verify the last part of Thm.0.0.2 pp.18-19, that EaX = aEX.
3. classical expectation was ever the modern object
Suppose, as in question 3 of Chpt.2, p.57 (but here in slightly di↵erent notation), for any X ≥ 0, we simply define
thus to find that, therefore, EX is actually the riemann integral
a) Show that ES = ES S ∈ S.
b) Verify as well that 0 ≤ X ≤ Y 0 ≤ EX ≤ EY.
c) Hence demonstrate how it is that for any X ∈ R+ we will automatically come to the fully modern result
4. lebesgue’s linear space L
a) Prove that for X, Y ∈ L : X ≤ Y EX ≤ EY .
b) Prove that |X| ≤ Y, Y ∈ L X ∈ L.
c) Show that ||X|| = E|X| defines a (pseudo)-norm on L.
d) Verify that |EX| ≤ E|X|, and describe the circumstances for equality.
5. lebesgue’s linear space L, (cont’d)
a) Prove that X ∈ L E|X|I(|X| > n) → 0 as n → 1.
b) Prove that X ∈ L nP(|X| > n) → 0 as n → 1, but the converse is false.
[Note: It might be useful to know that the derivative of lnln x is 1/x ln x.]
6. R-valued functions of abstract random variables
given
a)
b)
7. SLLN (L2 version): (Xi, i ∈ N) IID X w.
a) Prove that for Z ≥ 0, we do indeed have it that
b) Hence verify the identity