Difference between revisions of "Lebesgue measure"

Revision as of 18:33, 15 March 2015

The set function

$\lambda^n:(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))\rightarrow\mathbb{R}_{\ge}$ ^[1] that assigns every half-open rectangle

$[[a,b))=[a_1,b_1)\times\cdots\times[a_n,b_n)\in\mathcal{J}$ as follows:

$\lambda^n\Big([[a,b))\Big)=\Pi^n_{i=1}(b_i-a_i)$

Where

$\mathcal{J}=$ the set of all half-open-half-closed 'rectangles' in

$\mathbb{R}^n$

Note that it can be shown

$\mathcal{B}(\mathbb{R}^n)=\sigma(\mathcal{J})$ where

$\sigma(\mathcal{J})$ is the

$\mathcal{J}$

@@ Line 8: / Line 8: @@
 Where <math>\mathcal{J}=</math> the set of all half-open-half-closed 'rectangles' in <math>\mathbb{R}^n</math>
-Note that it can be shown <math>\mathcal{B}(\mathbb{R}^n)=\sigma(\mathcal{J})</math> where <math>\sigma(\mathcal{J})</math> is the [[Sigma-algebra|{{Sigma|algebra}} [[Sigma-algebra generated by|generated by]] <math>\mathcal{J}</math>
+Note that it can be shown <math>\mathcal{B}(\mathbb{R}^n)=\sigma(\mathcal{J})</math> where <math>\sigma(\mathcal{J})</math> is the [[Sigma-algebra|{{Sigma|algebra}}]] [[Sigma-algebra generated by|generated by]] <math>\mathcal{J}</math>
 {{Definition|Measure Theory}}
 ==References==