Difference between revisions of "Lebesgue measure"
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<math>\lambda^n\Big([[a,b))\Big)=\Pi^n_{i=1}(b_i-a_i)</math> | <math>\lambda^n\Big([[a,b))\Big)=\Pi^n_{i=1}(b_i-a_i)</math> | ||
| + | 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> | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
==References== | ==References== | ||
Revision as of 18:32, 15 March 2015
Definition
The set function λn:(Rn,B(Rn))→R≥[1] that assigns every half-open rectangle [[a,b))=[a1,b1)×⋯×[an,bn)∈J as follows:
λn([[a,b)))=Πni=1(bi−ai)
Where J= the set of all half-open-half-closed 'rectangles' in Rn
Note that it can be shown B(Rn)=σ(J) where σ(J) is the [[Sigma-algebra|σ-algebra generated by J
References
- Jump up ↑ P27 - Measures, Integrals and Martingales - Rene L. Schilling
