Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<strong>1.</strong>6. MEASURABILITY AND LEBESGUE INTEGRAL<br />
The next theorem says that for any measurable function f on<br />
IR, we can always find a sequence of measurable functions that<br />
converge to f. This will allow the definition of the Lebesgue integral.<br />
Theorem <strong>1.</strong>4 Let f : Ω → IR. It holds:<br />
(a) f isapositivemeasurablefunctionif<strong>and</strong>onlyiff = lim n→∞ f n ,<br />
where {f n } n∈IN is an increasing sequence of non negative simple<br />
functions.<br />
(b) f isameasurablefunctionif<strong>and</strong>onlyiff = lim n→∞ f n ,where<br />
{f n } n∈IN is an increasing sequence of simple functions.<br />
Definition <strong>1.</strong>28 (Lebesgueintegralfor non-negativefunctions)<br />
Letf beanon-negativemeasurablefunctiondefinedover(Ω,A,µ)<br />
<strong>and</strong> {f n } n∈IN be an increasing sequence of simple functions that<br />
converge pointwise to f (this sequence can be always constructed).<br />
The Lebesgue integral of f with respect to the σ-finite measure µ<br />
is defined as ∫<br />
f dµ = lim f n dµ.<br />
Ω n→∞<br />
∫Ω<br />
Thepreviousdefinitioniscorrectduetothefollowinguniqueness<br />
theorem.<br />
Theorem <strong>1.</strong>5 (Uniqueness of Lebesgue integral)<br />
Let f be a non-negative measurable function. Let {f n } n∈IN <strong>and</strong><br />
{g n } n∈IN be two increasing sequences of non-negative simple functions<br />
that converge pointwise to f. Then<br />
∫<br />
lim<br />
n→∞<br />
Ω<br />
f n dµ = lim g n dµ.<br />
n→∞<br />
∫Ω<br />
ISABEL MOLINA 26