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"
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==Statement== | ==Statement== | ||
| − | </noinclude>Suppose that {{M|\mu}} is either a [[measure]] (or a [[pre-measure]] | + | </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]]'' [[sequence|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)}} | ** {{M|A\subseteq\bigcup_i A_i\implies\mu(A)\le\sum_{i}\mu(A_i)}} | ||
Latest revision as of 20:44, 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
