Difference between revisions of "The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set/Statement"
From Maths
m (Alec moved page The (pre-)measure of a set is less than the sum of the (pre-)measures of the elements of a covering for that set/Statement to [[The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering f...) |
m (Adding note about sub-additivity and difference.) |
||
| Line 1: | Line 1: | ||
<noinclude> | <noinclude> | ||
==Statement== | ==Statement== | ||
| − | </noinclude>Suppose that {{M|\mu}} is either a [[measure]] or a [[pre-measure]] on the [[sigma-ring|{{sigma|ring}}]] or [[ring of sets|ring]], {{M|\mathcal{R} }} then{{rMTH}}: | + | </noinclude>Suppose that {{M|\mu}} is either a [[measure]] (or a [[pre-measure]]( on the [[sigma-ring|{{sigma|ring}}]] (or [[ring of sets|ring]]), {{M|\mathcal{R} }} then{{rMTH}}: |
| − | * for all {{M|A\in\mathcal{R} }} and for all [[countably infinite]] or [[finite]] sequences {{M|(A_i)\subseteq\mathcal{R} }} we have: | + | * for all {{M|A\in\mathcal{R} }} and for all ''[[countably infinite]]'' or ''[[finite]]'' [[sequence|sequences]] {{M|(A_i)\subseteq\mathcal{R} }} we have: |
| − | ** {{M|A\subseteq\bigcup_i A_i\implies\mu(A)\le\sum_{i}\mu(A_i)}}<noinclude> | + | ** {{M|A\subseteq\bigcup_i A_i\implies\mu(A)\le\sum_{i}\mu(A_i)}} |
| + | '''Note: ''' this is slightly different to [[sigma-subadditivity]] (or [[subadditivity]]) which states that {{M|1=\mu\left(\bigcup_i A_i\right)\le\sum_i\mu(A_i)}} (for a pre-measure, we would require {{M|\bigcup_i A_i\in\mathcal{R} }} which isn't guaranteed for countably infinite sequences)<noinclude> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Theorem Of|Measure Theory}} | {{Theorem Of|Measure Theory}} | ||
</noinclude> | </noinclude> | ||
Revision as of 20:27, 31 July 2016
Statement
Suppose that [ilmath]\mu[/ilmath] is either a measure (or a pre-measure( on the [ilmath]\sigma[/ilmath]-ring (or ring), [ilmath]\mathcal{R} [/ilmath] then[1]:
- for all [ilmath]A\in\mathcal{R} [/ilmath] and for all countably infinite or finite sequences [ilmath](A_i)\subseteq\mathcal{R} [/ilmath] we have:
- [ilmath]A\subseteq\bigcup_i A_i\implies\mu(A)\le\sum_{i}\mu(A_i)[/ilmath]
Note: this is slightly different to sigma-subadditivity (or subadditivity) which states that [ilmath]\mu\left(\bigcup_i A_i\right)\le\sum_i\mu(A_i)[/ilmath] (for a pre-measure, we would require [ilmath]\bigcup_i A_i\in\mathcal{R} [/ilmath] which isn't guaranteed for countably infinite sequences)
References
