Slides Chapter 1. Measure Theory and Probability

1.6. MEASURABILITY AND LEBESGUE INTEGRAL
(b) If {A n } n∈IN is a sequence of disjoint sets with A =
then ∫ ∞
A f dµ = ∑
∫
n=1
A n
f dµ.
∞⋃
A n ,
(c) If two measurable functions f and g are equal in all parts of
their domain except for a subset with measure µ zero and f
is Lebesgue integrable, then g is also Lebesgue integrable and
their Lebesgue integral is the same, that is,
∫ ∫
If µ({ω ∈ Ω : f(ω) ≠ g(ω)}) = 0, then f dµ = gdµ.
(d) Linearity: If f and g are Lebesgue integrable functions and a
and b are real numbers, then
∫ ∫ ∫
(af +bg)dµ = a f dµ+b gdµ.
A
(e) Monotonocity: If f and g are Lebesgue integrable and f < g,
then ∫ ∫
f dµ ≤ gdµ.
Theorem 1.7 (Monotone convergence theorem)
Consider a point-wise non-decreasing sequence of [0,∞]-valued
measurable functions {f n } n∈IN (i.e., 0 ≤ f n (x) ≤ f n+1 (x), ∀x ∈
IR, ∀n > 1) with lim n→∞ f n = f. Then,
∫ ∫
lim
n→∞
Ω
f n dµ =
A
Ω
fdµ.
Theorem 1.8 (Dominated convergence theorem)
Consider a sequence of real-valued measurable functions {f n } n∈IN
with lim n→∞ f n = f. Assume that the sequence is dominated
Ω
A
n=1
Ω
ISABEL MOLINA 29