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{{Requires references|grade=C|msg=I've sort of forged my own path with regards to what we call a pre-measure. This is because I have found it easier to consider a pre-measure as a very similar thing to a [[measure]], and then shown what other authors call pre-measures extend uniquely to my kind of pre-measure. They also call my kind of pre-measure a pre-measure.}}
==Definition==
<div style="display:none;">{{Extra Maths}}</div>Given a [[ring of sets]], {{M|\mathcal{R} }}, a ''(non-negative) pre-measure'' is an ''[[extended real valued]]'' [[countably additive set function]], {{M|\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} with the following properties:
# {{M|1=\bar{\mu}(\emptyset)=0}}
# {{M|1=\forall(A_n)_{n=1}^\infty\subseteq\mathcal{R} }}[[pairwise disjoint|{{M|\text{ pairwise disjoint} }}]]{{MM|1=\left[\left(\bigudot_{n=1}^\infty A_n\right)\in\mathcal{R}\implies\left(\bar{\mu}\left(\bigudot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\bar{\mu}(A_n)\right)\right]}}<ref group="Note">There is a slight abuse of notation here, by the nature of [[implies]] if the LHS is false, we do not care if the RHS is true or false! However if the LHS is false here, the RHS doesn't even make sense (as {{M|\bar{\mu}(A)}} makes no sense when {{M|A}} is not in the domain of {{M|\bar{\mu} }})</ref><ref group="Note">By using {{M|\bigudot}} we are making it clear that the sequence is that of pairwise disjoint sets and going forward we shall not write "pairwise disjoint" when the [[union]] symbol implies it. For clarity the full statement is thus:
* {{MM|1=\forall(A_n)_{n=1}^\infty\subseteq\mathcal{R}\left[\underbrace{\left(\forall i,j\in\mathbb{N}[i\ne j\implies A_i\cap A_j=\emptyset]\right)}_{\text{the }S_n\text{ are pairwise disjoint} }\overbrace{\wedge}^\text{and}\left(\left(\bigudot_{n=1}^\infty A_n\right)\in\mathcal{R}\implies\left(\bar{\mu}\left(\bigudot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\bar{\mu}(A_n)\right)\right)\right]}}</ref>
{{Caution|Be aware that this convention differs from what many authors define as a pre-measure (although it meets their definition) see [[#Conventions and terminology|conventions and terminology]] below}}
==Conventions and terminology==

==See also==
* [[Pre-measurable space]]
* [[Measure]] - which we obtain by:
** [[Extending pre-measures to outer-measures]] as a part of
** [[Extending pre-measures to measures]]
* [[Semi-ring of sets]]
** [[Pre-measure on a semi-ring]]
* [[Types of set algebras]]

==Notes==
<references group="Note"/>
==References==
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