Slides Chapter 1. Measure Theory and Probability

1.7. DISTRIBUTION FUNCTION
Definition 1.36 (Probability mass function)
The collection of numbers P F ({a 1 }),...,P F ({a n }),..., such that
P F ({a i }) > 0, ∀i and ∑ ∞
i=1 P F({a i }) = 1, is called probability
mass function.
Remark 1.5 Observe that
F(x) = P F (−∞,x] = ∑ a i ≤xP F ({a i }).
Thus, F(x) is a step function and the length of the step at a n is
exactly the probability of a n , that is,
P F ({a i }) = P(−∞,a n ]−P(−∞,a n ) = F(a n )− lim
x↓an
F(x)
= F(a n )−F(a n −).
Theorem 1.12 (Radon-Nykodym Theorem)
Givenameasurablespace(Ω,A), ifaσ-finitemeasureµon(Ω,A)
is absolutely continuous with respect to a σ-finite measure λ on
(Ω,A), then there is a measurable function f : Ω → [0,∞), such
that for any measurable set A,
∫
µ(A) = f dλ.
The function f satisfying the above equality is uniquely defined
up to a set with measure µ zero, that is, if g is another function
which satisfies the same property, then f = g except in a set with
measure µ zero. f is commonly written dµ/dλ and is called the
Radon–Nikodym derivative. Thechoiceofnotationandthename
of the function reflects the fact that the function is analogous to
a derivative in calculus in the sense that it describes the rate of
change of density of one measure with respect to another.
A
ISABEL MOLINA 33