A premeasure on a semiring may be extended uniquely to a premeasure on a ring
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Demote to grade A once fleshed out and grade C once (most of) a proof has been added
Statement
Given a premeasure on a semiring, [ilmath]\tilde{\mu}:\mathcal{F}\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath] (that is a function whose domain is a semiring of sets that is countably additive with [ilmath]\tilde{\mu}(\emptyset)=0[/ilmath]) then we may extend [ilmath]\srmu[/ilmath] to a premeasure, [ilmath]\rmu:R(\mathcal{F})\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath]^{[Note 1]}; furthermore this extension is unique^{[1]}. The details follow:
 The ring generated by a semiring is exactly the set of all finite disjoint unions of elements from that semiring.
 That is to say, [ilmath]R(\mathcal{F})=\left\{\left.\bigudot_{i=1}^nA_i\ \right\vert\ (A_i)_{i=1}^n\subseteq\mathcal{F}\right\}[/ilmath]
 so any [ilmath]A\in R(\mathcal{F}) [/ilmath] can be written as [ilmath]A=\bigudot_{i=1}^n A_i[/ilmath] for some finite sequence of pariwise disjoint sets, [ilmath] ({ A_i })_{ i = 1 }^{ n }\subseteq \mathcal{F} [/ilmath]^{[Note 2]}
 We define the induced premeasure, [ilmath]\rmu:R(\mathcal{F})\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath] as follows:
 [ilmath]\rmu:\bigudot_{i=1}^nA_i\mapsto\sum_{i=1}^n\srmu(A_i)[/ilmath], and we claim this map is welldefined
Prerequisites
Proof
Grade: A
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I need to create the premeasure on a semiring page and the ring of sets generated by a semiring is the set containing the semiring and all finite disjoint unions page before proceeding. See page 39 in^{[1]}
See also
Notes
 ↑ Here [ilmath]R(X)[/ilmath] denotes the ring of sets generated by a collection of sets, [ilmath]X[/ilmath].
 ↑ I've mentioned it a few times but in case it isn't clear:
 For [ilmath]A\in R(\mathcal{F})[/ilmath] we have [ilmath]A=\bigudot_{i=1}^nA_i[/ilmath] for some finite sequence, [ilmath] ({ A_i })_{ i = 1 }^{ n }\subseteq \mathcal{F} [/ilmath], note the elements of the sequence are in [ilmath]\mathcal{F} [/ilmath]
References
 ↑ ^{1.0} ^{1.1} Measures, Integrals and Martingales  René L. Schilling
