View source for 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==
</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:
** {{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/>
{{Theorem Of|Measure Theory}}
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