Slides Chapter 1. Measure Theory and Probability

1.7. DISTRIBUTION FUNCTION
Proposition 1.13 (Norm in L p space)
The function φ : L p (µ) → IR that assigns to each function f ∈
L p (µ) the value φ(f) = (∫ Ω |f|p dµ ) 1/p
is a norm in the vector
space L p (µ) and it is denoted as
(∫ 1/p
‖f‖ p = |f| dµ) p .
Now we can introduce a metric in L p (µ) as
Ω
d(f,g) = ‖f −g‖ p .
A vector space with a metric obtained from a norm is called a
metric space.
Problems
1. Proof that if f and g are measurable, then max{f,g} and
min{f,g} are measurable.
1.7 Distribution function
Wewillconsider theprobability space (IR,B,P). The distribution
function will be a very important tool since it will summarize the
probabilities over Borel subsets.
Definition 1.33 (Distribucion function)
Let (IR,B,P) a probability space. The distribution function
(d.f.) associated with the probability function P is defined as
F : IR → [0,1]
x F(x) = P(−∞,x].
WecanalsodefineF(−∞) = lim F(x)andF(+∞) = lim F(x).
x↓−∞ x↑+∞
Then, the distribution function is F : [−∞,+∞] → [0,1].
ISABEL MOLINA 31