Measure, Integration & Real Analysis, 2021a

Chapter 12 Probability Measures 393
Your understanding of the proof of the next result should be enhanced by Exercise
13, which asserts that if the function H : R → (0, 1) appearing in the next result is
continuous and injective, then the random variable X : (0, 1) → R in the proof is the
inverse function of H.
12.29 characterization of distribution functions
Suppose H : R → [0, 1] is a function. Then there exists a probability space
(Ω, F, P) and a random variable X on (Ω, F) such that H = ˜X if and only if
the following conditions are all satisfied:
(a) s < t ⇒ H(s) ≤ H(t) (in other words, H is an increasing function);
(b)
(c)
lim H(t) =0;
t→−∞
lim H(t) =1;
t→∞
(d) lim
t↓s
H(t) =H(s) for every s ∈ R (in other words, H is right continuous).
Proof First suppose H = ˜X for some probability space (Ω, F, P) and some random
variable X on (Ω, F). Then (a) holds because s < t implies (−∞, s] ⊂ (−∞, t].
Also, (b) and (d) follow from 2.60. Furthermore, (c) follows from 2.59, completing
the proof in this direction.
To prove the other direction, now suppose that H satisfies (a) through (d). Let
Ω =(0, 1), let F be the collection of Borel subsets of the interval (0, 1), and let P
be Lebesgue measure on F. Define a random variable X by
12.30 X(ω) =sup{t ∈ R : H(t) < ω}
for ω ∈ (0, 1). Clearly X is an increasing function and thus is measurable (in other
words, X is indeed a random variable).
Suppose s ∈ R. Ifω ∈ ( 0, H(s) ] , then
X(ω) ≤ X ( H(s) ) = sup{t ∈ R : H(t) < H(s)} ≤s,
where the first inequality holds because X is an increasing function and the last
inequality holds because H is an increasing function. Hence
( ]
12.31
0, H(s) ⊂{X ≤ s}.
If ω ∈ (0, 1) and X(ω) ≤ s, then H(t) ≥ ω for all t > s (by 12.30). Thus
H(s) =lim H(t) ≥ ω,
t↓s
where the equality above comes from (d). Rewriting the inequality above, we have
ω ∈ ( 0, H(s) ] . Thus we have shown that {X ≤ s} ⊂ ( 0, H(s) ] , which when
combined with 12.31 shows that {X ≤ s} = ( 0, H(s) ] . Hence
as desired.
˜X(s) =P(X ≤ s) =P( (0, H(s)
] ) = H(s),
Measure, Integration & Real Analysis, by Sheldon Axler