Slides Chapter 1. Measure Theory and Probability

1.6. MEASURABILITY AND LEBESGUE INTEGRAL
Remark 1.2 The Lebesgue integral of a measurable function f
defined from a measurable space (Ω,A) on (IR,B), over a Borel
set I = (a,b) ∈ B, will also be expressed as
∫ ∫ b
f dµ = f(x)dµ(x).
I
a
Proposition 1.11 If a function f : IR → IR + = [0,∞) is Riemann
integrable, then it is also Lebesgue integrable (with respect
to the Lebesgue measure λ) and the two integrals coincide, i.e.,
∫ ∫
f(x)dµ(x) = f(x)dx.
A
Example 1.16 The Dirichlet function 1 lQ is not continuous in
any point of its domain.
• This function is not Riemann integrable in [0,1] because each
subinterval will contain at least a rational number and and
irrational number, since both sets are dense in IR. Then, each
superior sum is 1 and also the infimum of this superior sums,
whereas each lowersum is 0, the same as the suppremum of all
lowersums. Sincethesuppresumandtheinfimumaredifferent,
then the Riemann integral does not exist.
• However, it is Lebesgue integrable on [0,1] with respect to the
Lebesgue measure λ, since by definition
∫
1 lQ dλ = λ(lQ∩[0,1]) = 0,
[0,1]
since lQ is numerable.
Proposition 1.12 (Properties of Lebesgue integral)
(a) If P(A) = 0, then ∫ Af dµ = 0.
A
ISABEL MOLINA 28