Slides Chapter 1. Measure Theory and Probability

1.6. MEASURABILITY AND LEBESGUE INTEGRAL
The next theorem says that for any measurable function f on
IR, we can always find a sequence of measurable functions that
converge to f. This will allow the definition of the Lebesgue integral.
Theorem 1.4 Let f : Ω → IR. It holds:
(a) f isapositivemeasurablefunctionifandonlyiff = lim n→∞ f n ,
where {f n } n∈IN is an increasing sequence of non negative simple
functions.
(b) f isameasurablefunctionifandonlyiff = lim n→∞ f n ,where
{f n } n∈IN is an increasing sequence of simple functions.
Definition 1.28 (Lebesgueintegralfor non-negativefunctions)
Letf beanon-negativemeasurablefunctiondefinedover(Ω,A,µ)
and {f n } n∈IN be an increasing sequence of simple functions that
converge pointwise to f (this sequence can be always constructed).
The Lebesgue integral of f with respect to the σ-finite measure µ
is defined as ∫
f dµ = lim f n dµ.
Ω n→∞
∫Ω
Thepreviousdefinitioniscorrectduetothefollowinguniqueness
theorem.
Theorem 1.5 (Uniqueness of Lebesgue integral)
Let f be a non-negative measurable function. Let {f n } n∈IN and
{g n } n∈IN be two increasing sequences of non-negative simple functions
that converge pointwise to f. Then
∫
lim
n→∞
Ω
f n dµ = lim g n dµ.
n→∞
∫Ω
ISABEL MOLINA 26