1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
|
---
title: Lebesgue Integral
parent: Measure and Integration
nav_order: 2
has_children: true
has_toc: false
---
# {{ page.title }}
For this entire section we fix a measure space $(X,\mathcal{A},\mu)$.
## Integration of Nonnegative Step Functions
{% definition %}
Let $f : X \to \RR$ be a nonnegative step function
with representation $f = \sum_{i=1}^n \alpha_i \chi_{A_i}$,
where $\alpha_1, \ldots, \alpha_n \ge 0$ and
$A_1, \ldots, A_n \in \mathcal{A}$.
We define the *integral of $f$ on $X$ with respect to $\mu$* by
$$
\int_X f \, d\mu = \sum_{i=1}^n \alpha_i \, \mu(A_i) \in [0,\infty].
$$
{% enddefinition %}
TODO: This does not depend on the representation of $f$.
## Integration of Nonnegative Measurable Functions
{% theorem Approximation by Step Functions %}
Every nonnegative measurable function $f : X \to \overline{\RR}$
is the pointwise limit of an increasing sequence $(s_n)$ of
nonnegative step functions $s_n : X \to \RR$.
{% endtheorem %}
{% definition %}
Let $f : X \to \overline{\RR}$ be a nonnegative measurable function
and let $(s_n)$ be a sequence of nonnegative step functions
with $s_n \uparrow f$.
We define the *integral of $f$ on $X$ with respect to $\mu$* by
$$
\int_X f \, d\mu = \lim_{n \to \infty} \int_X s_n \, d\mu \in [0,\infty].
$$
{% enddefinition %}
## Integrable Functions
Recall that the positive and (flipped) negative parts
of a function $f : X \to \overline{R}$ are defined by
$$
f^+ = \max(f,0) \qquad
f^- = \max(-f,0),
$$
and that $f$ is measurable if and only if both $f^+$ and $f^-$ are measurable.
We have $f = f^+\! - f^-$.
{% definition Integrable Function, Lebesgue Integral %}
A measurable function $f : X \to \overline{\RR}$ is said to be
*integrable on $X$ with respect to $\mu$* if the integrals
$$
\int_X f^+ \, d\mu, \qquad \int_X f^- \, d\mu
\tag{$*$}
$$
are both finite.
In this case the *(Lebesgue) integral of $f$ on $X$ with respect to $\mu$* is defined as
$$
\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \RR.
$$
{% enddefinition %}
Sometimes it is convenient to have a slightly more general notion of integrability:
{% definition Quasi-Integrable Function %}
A measurable function $f : X \to \overline{\RR}$ is said to be
*quasi-integrable on $X$ with respect to $\mu$* if at least one of the integrals
$(*)$ is finite.
In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as
$$
\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \overline{\RR}.
$$
{% enddefinition %}
{% definition %}
A measurable function $f : X \to \CC$ is said to be
*integrable on $X$ with respect to $\mu$* if
$\Re f$ and $\Im f$ are integrable on $X$ with respect to $\mu$.
In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as
$$
\int_X f \, d\mu = \int_X \Re f \, d\mu + i \int_X \Im f \, d\mu \in \CC.
$$
{% enddefinition %}
## Integration on Measurable Subsets
{% definition %}
For any measurable subset $A \subset X$ we define
the *integral on $A$* of a (quasi-)integrable function $f : X \to \overline{\RR}$ (or $\CC$) by
$$
\int_A f \, d\mu =
\int_X \chi_A f \, d\mu.
$$
{% enddefinition %}
|
