Slides Chapter 1. Measure Theory and Probability

1.6. MEASURABILITY AND LEBESGUE INTEGRAL
by
∫
an integrable function g (i.e., |f n (x)| ≤ g(x), ∀x ∈ IR, with
Ωg(x)dµ < ∞). Then,
∫ ∫
f n dµ = fdµ.
lim
n→∞
Ω
Theorem 1.9 (Hölder’s inequality)
Let (Ω,A,µ) be a measure space. Let p,q ∈ IR such that p > 1
and 1/p+1/q = 1. Let f and g be measurable functions with |f| p
and |g| q µ-integrable (i.e., ∫ |f| p dµ < ∞ and ∫ |g| q dµ < ∞.).
Then, |fg| is also µ-integrable (i.e., ∫ |fg|dµ < ∞) and
∫ (∫ ) 1/p (∫ 1/q
|fg|dµ ≤ |f| p dµ |g| dµ) q .
Ω
Ω
The particular case with p = q = 2 is known as Schwartz’s inequality.
Theorem 1.10 (Minkowski’s inequality)
Let (Ω,A,µ) be a measure space. Let p ≥ 1. Let f and g be
measurablefunctionswith|f| p and|g| p µ-integrable. Then,|f+g| p
is also µ-integrable and
(∫ 1/p (∫ 1/p (∫ 1/p
|f +g| dµ) p ≤ |f| dµ) p + |g| dµ) p .
Ω
Ω
Definition 1.32 (L p space)
Let (Ω,A,µ) be a measure space. Let p ≠ 0. We define the L p (µ)
space as the set of measurable functions f with |f| p µ-integrable,
that is,
L p (µ) = L p (Ω,A,µ) =
Ω
Ω
{
f : f measurable and
Ω
∫
Ω
}
|f| p dµ < ∞ .
BytheMinkowski’sinequality,theL p (µ)spacewith1 ≤ p < ∞
is a vector space in IR, in which we can define a norm and the
corresponding metric associated with that norm.
ISABEL MOLINA 30