Slides Chapter 1. Measure Theory and Probability

1.7. DISTRIBUTION FUNCTION
Proposition 1.14 (Properties of the d.f.)
(i) The d.f. is monotone increasing, that is,
(ii) F(−∞) = 0 and F(+∞) = 1.
x < y ⇒ F(x) ≤ F(y).
(iii) F is right continuous for all x ∈ IR.
Remark 1.3 If the d.f. was defined as F(x) = P(−∞,x), then
it would be left continuous.
We can speak about a d.f. without reference to the probability
measure P that is used to define the d.f.
Definition 1.34 A function F : [−∞,+∞] → [0,1] is a d.f. if
and only if satisfies Properties (i)-(iii).
Now, given a d.f. F verifying (i)-(iii), is there a unique probability
function over (IR,B) whose d.f. is exactly F?
Proposition 1.15 Let F : [−∞,+∞] → [0,1] be a function
thatsatisfiesproperties(i)-(iii). Then,thereisauniqueprobability
functionP F definedover(IR,B)suchthatthedistributionfunction
associated with P F is exactly F.
Remark 1.4 Leta,bberealnumberswitha < b. ThenP(a,b] =
F(b)−F(a).
Theorem 1.11 The set D(F) of discontinuity points of F is finite
or countable.
Definition 1.35 (Discrete d.f.)
Ad.f. F isdiscreteifthereexistafiniteorcountableset{a 1 ,...,a n ,...} ⊂
IR such that P F ({a i }) > 0, ∀i and ∑ ∞
i=1 P F({a i }) = 1, where P F
is the probability function associated with F.
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