Real-valued Measurable Functions
To discuss integration, we have to focus on functions with values in the extended real numbers.
Proposition
A function
on a measure space with values in the extended real numbers (with standard Borel sigma algebra) is measurable if and only if for all .
Integrable Functions
Suppose
is a measure space. A measurable function is integrable if
Properties of Integration
The elementary properties of integral continue to hold:
Proposition
The following properties hold for integrals on a measure space
:
- Linearity:
- Additivity: If
and are disjoint subsets of , then - Monotonicity: If
, then - Triangle inequality:
The basic limit theorems also hold:
Fatou's Lemma
If
is a sequence of non-negative measurable functions on , then
Monotone Convergence Theorem
If
is a sequence of non-negative measurable functions with on , then
Bounded Convergence Theorem
If
is a sequence of measurable functions on with and for all . If a.e., then
Dominated Convergence Theorem
If
is a sequence of measurable functions with pointwise a.e. on , and for some integrable , then
The Space of Integrable Functions
We denote by
the space of all integrable functions on , modulo the equivalence relation of being equal almost everywhere, with the norm defined by the integral of the absolute value:
Proposition
The space
is a complete normed vector space.
We define
to be the equivalence classes of measurable functions for which . It posses an inner product: which induces a norm
Proposition
The space
is a (possibly non-separable) Hilbert space.
Proposition
Suppose
is a measure space. Then any for is integrable if .
Proof This follows from the Cauchy-Schwarz inequality:
Product Measures and Fubini’s Theorem
Measurable Rectangle
Let
and be measurable spaces. A measurable rectangle is a set of the form , where and .
Lemma
Let
and be measurable spaces. Then is a -algebra on . It is the smallest -algebra containing all measurable rectangles.
Product Measure
Suppose
and are two measure spaces. A measure on is called a product measure if for all measurable rectangles , we have
Product Measure Theorem
Suppose
and are two measure spaces. Then there exists a measure on such that for all measurable rectangles . Moreover, if and are -finite, then is unique and -finite, and we denote it as .
Proof
Tonelli's Theorem
Suppose
and are two -finite measure spaces, and is -measurable. Then
is an -measurable function on , is an -measurable function on , and
Fubini's Theorem
Suppose
and are two -finite measure spaces, and is -integrable. Then is -integrable for a.e. , and is -integrable for a.e. . Moreover, we have
is an -measurable function on , is an -measurable function on , and