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"
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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 to [[The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that se...) |
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{{Stub page|I could probably write the statement a bit better}} | {{Stub page|I could probably write the statement a bit better}} | ||
| − | ==[[The (pre-)measure of a set is | + | ==[[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|Statement]]== |
| − | {{:The (pre-)measure of a set is | + | {{: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}} |
==Proof== | ==Proof== | ||
{{Requires proof|See outline}} | {{Requires proof|See outline}} | ||
Revision as of 22:20, 8 April 2016
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I could probably write the statement a bit better
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)
Proof
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See outline
Outline:
- Construct a new sequence, [ilmath](B_i)[/ilmath] where the elements are pairwise disjoint
- Define [ilmath]B[/ilmath] as the union of all [ilmath](B_i)[/ilmath]
- As [ilmath]A\subseteq B[/ilmath] we can use the monotonicity of a measure/pre-measure.
References
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