Slides Chapter 1. Measure Theory and Probability

1.6. MEASURABILITY AND LEBESGUE INTEGRAL
Definition 1.29 (Lebesgueintegralfor generalfunctions)
For a measurable function f that can take negative values, we can
write it as the sum of two non-negative functions in the form:
f = f + −f − ,
wheref + (ω) = max{f(ω),0}isthepositive part off andf − (ω) =
max{−f(ω),0} is the negative part of f. If the integrals of f +
and f − are finite, then the Lebesgue integral of f is
∫ ∫ ∫
f dµ = f + dµ− f − dµ,
Ω
Ω
assuming that at least one of the integrals on the right is finite.
Definition 1.30 TheLebesgue integral ofameasurablefunction
f over a subset A ∈ A is defined as
∫ ∫
f dµ = f1 A dµ.
A
Definition 1.31 A function is said to be Lebesgue integrable if
and only if ∣∫
∣∣∣ f dµ
∣ < ∞.
Ω
Moreover, if instead of doing the decomposition f = f + − f −
we do another decomposition, the result is the same.
Theorem 1.6 Letf 1 ,f 2 ,g 1 ,g 2 benon-negative measurable functions
and let f = f 1 −f 2 = g 1 −g 2 . Then,
∫ ∫ ∫ ∫
f 1 dµ− f 2 dµ = g 1 dµ− g 2 dµ.
Ω
Ω
Proposition 1.10 A measurable function f is Lebesgue integrable
if and only if |f| is Lebesgue integrable.
Ω
Ω
Ω
Ω
ISABEL MOLINA 27