Uncertainty Theory
Baoding Liu @ Tsinghua University
Alegebra
Let Γ be a nonempty set (sometimes called universal set). A collection L
is called an algebra over Γ if the following three conditions hold:
(a) Γ ∈ L;
(b) if Λ ∈ L, then Λc ∈ L;
(c) if Λi ∈ L for i = 1, 2, · · · , n, then
n
[
Λi ∈ L.
i=1
σ-Algebra
If the condition (c) is replaced with closure under countable union, i.e.,
∞
[
Λi ∈ L,
i=1
then L is called a σ-algebra over Γ
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Example 1: The collection {∅, Γ} is the smallest σ-algebra over Γ, and
the power set (all subsets) is the largest σ-algebra.
Example 2: The collection {∅, Λ, Λc , Γ} is a σ-algebra over Γ.
Example 3: Let L be the collection of all finite disjoint unions of all
intervals of the form
(−∞, a],
(a, b],
(b, ∞),
∅.
Then L is an algebra over <, but not a σ-algebra because
∞
[
(0, (i − 1)/i] = (0, 1) 6∈ L.
i=1
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http://orsc.edu.cn/liu/ut.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
Measurable Space and Measurable Set
Definition
Let Γ be a nonempty set, and L a σ-algebra over Γ. Then (Γ, L) is called
a measurable space, and any element in L is called a measurable set.
Example: Let < be the set of real numbers.
Then (<, {∅, <}) is a measurable space in which there exist only two
measurable sets, one is ∅ and another is <.
Keep in mind that the intervals like [0, 1] and (0, +∞) are not measurable!
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Uncertainty Theory
Baoding Liu @ Tsinghua University
Product σ-Algebra
Definition
Let Li be σ-algebras over Γi , i = 1, 2, · · · , n, respectively. A measurable
rectangle in Γ is a set
Λ = Λ1 × Λ2 × · · · × Λn
where Λi ∈ Li for i = 1, 2, · · · , n. The smallest σ-algebra containing all
measurable rectangles of Γ is called the product σ-algebra, denoted by
L = L1 × L2 × · · · × Ln .
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http://orsc.edu.cn/liu/ut.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
Borel Algebra and Borel Set
Definition
The smallest σ-algebra B containing all open intervals is called the Borel
algebra over <, and any elements in B are called a Borel set.
Example 1: It has been proved that interval, open set, closed set, rational
numbers, and irrational numbers are all Borel sets.
Example 2: Let [a] represent the set of all rational numbers plus a.
If a1 − a2 is not a rational number, then [a1 ] and [a2 ] are disjoint.
Thus < is divided into an infinite number of disjoint sets.
Let A be a new set containing precisely one element from those sets.
Then A is not a Borel set.
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http://orsc.edu.cn/liu/ut.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
Measurable Function
Definition
A real-valued function f on a measurable space (Γ, L) is said to be
measurable if and only if
f −1 (B) ∈ L
for any Borel set B.
Example 1: Any monotone function is measurable.
Example 2: Any continuous function is measurable.
Example 3: The characteristic function of a set Λ is measurable if Λ is a
measurable set; and is not measurable if Λ is not.
(
1, if x ∈ Λ
f (x) =
0, if x 6∈ Λ
[email protected]
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