= Measure Space =

<<TableOfContents>>

----



== 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 [[Analysis/SigmaAlgebra|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,∞) ∪ {∞}''.

The convention for arithmetic with ∞ in measure spaces is that:
 * ''x + ∞ = ∞'' for all ''x ∈ [0,∞]''
 * ''x * ∞ = ∞'' for all ''x ∈ (0,∞]''
 * ''0 * ∞ = 0''

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''
 * '''σ additivity''': additivity for finite and [[Analysis/Cardinality|countably]] infinite elements, but not infinite elements
   * in the finite case, requiring that each is disjoint (i.e. ''A,,i,, ∩ A,,j,, = Ø, i != j''): {{attachment:countable.svg}}
   * in the countable case: {{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.



----
CategoryRicottone