Slides Chapter 1. Measure Theory and Probability

1.6. MEASURABILITY AND LEBESGUE INTEGRAL
functions defined from a measurable space (Ω,A) on the measurable
space (IR,B), where B is the Borel σ-algebra. We consider
also a σ-finite measure µ.
Definition 1.25 (Indicator function)
Given S ∈ A, an indicator function, 1 S : Ω → IR, gives value 1 to
elements of S and 0 to the rest of elements:
{ 1, ω ∈ S;
1 S (ω) =
0, ω /∈ S.
Next we define simple functions, which are linear combinations
of indicator functions.
Definition 1.26 (Simple function)
Let (Ω,A,µ) be a measure space. Let a i be real numbers and
{S i } n i=1 disjoint elements of A. A simple function has the form
n∑
φ = a i 1 Si .
i=1
Proposition 1.9 Indicators and simple functions are measurable.
Definition 1.27 (Lebesgueintegralfor simplefunctions)
(i) The Lebesgue integral of a simple function φ with respect to
a σ-finite measure µ is defined as
∫ n∑
φdµ = a i µ(S i ).
Ω
(ii) The Lebesgue integral of φ with respect to µ over a subset
i=1
A ∈ A is ∫
A
φdµ =
∫
A
φ·1 A dµ =
n∑
a i µ(A∩S i ).
i=1
ISABEL MOLINA 25