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| A measure space is a triple of: | A measure space is a triple ''(Ω,'' 𝒜'', μ)'', composed of: |
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| * a measure ''μ :'' 𝒜 ''-> [0,∞]'' | * a '''measure''' ''μ :'' 𝒜 ''-> [0,∞]'' |
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As for a measure, this is a [[Analysis/Functions|map]] from 𝒜 to non-negative real numbers. It must satisfy two conditions: * ''μ(Ø) = 0'' * additivity * for a [[Analysis/Cardinality|countable]] set of ''n'' elements, requiring that each is disjoint (i.e. ''A,,i,, ∩ A,,j,, = Ø, i != j''): {{attachment:countable.svg}} * for an infinite set: {{attachment:infinite.svg}} It should be clear that these conditions are the reason for conditions 1 and 3 on the definition of a σ algebra. |
Measure Space
Contents
Introduction
To motivate a theory of measures:
"[I]n one dimension, the intervals A := [0, 1] and B := [0, 2] are in one-to-one correspondence (using the bijection x -> 2x from A to B), but of course B is twice as long as A. So one can disassemble A into an uncountable number of points and reassemble them to form a set of twice the length." (p3, An Introduction to Measure Theory, Terence Tao)
Description
A measure space is a triple (Ω, 𝒜, μ), composed of:
a sample space Ω
a σ algebra 𝒜
Note that the double (Ω, 𝒜 ) is sometimes called a measurable space.
a measure μ : 𝒜 -> [0,∞]
Note that [0,∞] is shorthand for the more formally correct [0,∞) ∪ {∞}.
As for a measure, this is a map from 𝒜 to non-negative real numbers. It must satisfy two conditions:
μ(Ø) = 0
- additivity
for a countable set of n elements, requiring that each is disjoint (i.e. Ai ∩ Aj = Ø, i != j):
for an infinite set:
It should be clear that these conditions are the reason for conditions 1 and 3 on the definition of a σ algebra.