Slides Chapter 1. Measure Theory and Probability

1.7. DISTRIBUTION FUNCTION
Theorem 1.13 A finite measure µ on Borel subsets of the real
line is absolutely continuous with respect to Lebesgue measure if
and only if the point function
F(x) = µ((−∞,x])
is a locally and absolutely continuous real function.
If µ is absolutely continuous, then the Radon-Nikodym derivative
of µ is equal almost everywhere to the derivative of F. Thus,
the absolutely continuous measures on IR n are precisely those that
have densities; as a special case, the absolutely continuous d.f.’s
are precisely the ones that have probability density functions.
Definition 1.37 (Absolutely continuous d.f.)
A d.f. is absolutely continuous if and only if there is a nonnegative
Lebesgue integrable function f such that
∫
∀x ∈ IR,F(x) = fdλ,
(−∞,x]
where λ is the Lebesgue measure. The function f is called probability
density function, p.d.f.
Proposition 1.16 Let f : IR → IR + = [0,∞) be a Riemann
integrable function such that ∫ +∞
∫ −∞
f(t)dt = 1. Then, F(x) =
x
−∞f(t)dt is an absolutely continuous d.f. whose associated p.d.f
is f.
All the p.d.f.’s that we are going to see are Riemann integrable.
Proposition 1.17 Let F be an absolutely continuous d.f. Then
it holds:
(a) F is continuous.
ISABEL MOLINA 34