Slides Chapter 1. Measure Theory and Probability

1.6. MEASURABILITY AND LEBESGUE INTEGRAL
1.6 Measurability and Lebesgue integral
A measurable function relates two measurable spaces, preserving
the structure of the events.
Definition 1.24 Let (Ω 1 ,A 1 ) and (Ω 2 ,A 2 ) be two measurable
spaces. A function f : Ω 1 → Ω 2 is said to be measurable if and
only if ∀B ∈ A 2 , f −1 (B) ∈ A 1 , where f −1 (B) = {ω ∈ Ω 1 :
f(ω) ∈ B}.
The sum, product, quotient (when the function in the denominatorisdifferentfromzero),maximum,minimumandcomposition
of two measurable functions is a measurable function. Moreover,
if {f n } n∈IN is a sequence of measurable functions, then
sup{f n }, inf {f n}, liminf f n, limsupf n , lim f n ,
n∈IN n∈IN n∈IN n∈IN n→∞
assuming that they exist, are also measurable. If they are infinite,
we can consider IR ¯ instead of IR.
The following result will gives us a tool useful to check if a
function f from (Ω 1 ,A 1 ) into (Ω 2 ,A 2 ) is measurable.
Theorem 1.3 Let (Ω 1 ,A 1 ) and (Ω 2 ,A 2 ) be measure spaces and
let f : Ω 1 → Ω 2 . Let C 2 ⊂ P(Ω 2 ) be a collection of subsets that
generates A 2 , i.e, such that σ(C 2 ) = A 2 . Then f is measurable if
and only of f −1 (A) ∈ A 1 , ∀A ∈ C 2 .
Corolary 1.1 Let (Ω,A) be a measure space and f : Ω → IR a
function. Then f is measurable if and only if f −1 (−∞,a] ∈ A,
∀a ∈ IR.
The Lebesgue integral is restricted to measurable functions. We
are going to define the integral by steps. We consider measurable
ISABEL MOLINA 24