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| − | An [[Additive function|additive function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808601.png" /> defined on a family of sets in a topological space whose total variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808602.png" /> (cf. [[Total variation of a function|Total variation of a function]]) satisfies the condition
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| | + | ''rule of infinite induction, $\omega$-rule'' |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808603.png" /></td> </tr></table>
| + | A [[Derivation rule|derivation rule]] stating that if for an arithmetic formula $\phi(x)$ the propositions $\phi(0),\phi(1),\ldots,$ have been proved, then the proposition $\forall x\phi(x)$ can be regarded as being proved. This rule was first brought into consideration by R. Carnap [[#References|[1]]]. Carnap's rule uses an infinite set of premises and is therefore inadmissible within the structure of the formal theories of D. Hilbert. The concept of a derivation in a system with the Carnap rule is undecidable. In mathematical logic one uses, for the study of formal arithmetic, the constructive Carnap rule: If there is an algorithm which for a natural number $n$ provides a derivation of the formula $\phi(n)$, then the proposition $\forall x\phi(x)$ can be regarded as being proved (the restricted $\omega$-rule, the rule of constructive infinite induction). Classical arithmetic calculus, which by Gödel's theorem is incomplete, becomes complete on adding the constructive Carnap rule (see [[#References|[2]]], [[#References|[3]]]). |
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808604.png" /> denotes the interior of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808606.png" /> the closure of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808607.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r0808609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086010.png" /> are in the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086011.png" />). Every bounded additive regular set function, defined on a semi-ring of sets in a compact topological space, is countably additive (Aleksandrov's theorem).
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| − | The property of regularity can also be related to a measure, as a special case of a set function, and one speaks of a regular measure, defined on a topological space. For example, the [[Lebesgue measure|Lebesgue measure]] is regular.
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| | ====References==== | | ====References==== |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Wiley (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.D. Aleksandrov, "Additive set-functions in abstract spaces" ''Mat. Sb.'' , '''9''' (1941) pp. 563–628 (In Russian)</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Carnap, "The logical syntax of language" , Kegan Paul, Trench & Truber (1937) (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Kuznetsov, ''Uspekhi Mat. Nauk'' , '''12''' : 4 (1957) pp. 218–219</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.R. Shoenfield, "On a restricted $\omega$-rule" ''Bull. Acad. Polon. Sci. Cl. III'' , '''7''' (1959) pp. 405–407</TD></TR></table> |
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| − | ====Comments==== | |
| − | Although a set function is called regular if it satisfies a property of approximation from below or above involving "nice" sets, the precise meaning of "regular" usually depends on the context (and on the author). For example, a (Carathéodory) [[Outer measure|outer measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086012.png" /> is called regular if for every part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086014.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086015.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086016.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086017.png" />-measurable set containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086018.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086019.png" /> is a topological space, the outer measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086020.png" /> is called Borel regular if Borel sets are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086021.png" />-measurable and if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086022.png" /> above can be taken Borel. On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086023.png" /> is a metrizable space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086024.png" /> is a finite measure on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086025.png" />-field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086026.png" /> is always regular in the sense of the article above. In this setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086027.png" /> is often called inner regular, or just regular, if for any Borel subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086028.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086029.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086030.png" /> a countable union of compact sets included in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086031.png" />, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086032.png" /> is a [[Radon measure|Radon measure]]. Instead of calling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080860/r08086033.png" /> Radon, one nowadays most often says that it is tight.
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rule of infinite induction, $\omega$-rule
A derivation rule stating that if for an arithmetic formula $\phi(x)$ the propositions $\phi(0),\phi(1),\ldots,$ have been proved, then the proposition $\forall x\phi(x)$ can be regarded as being proved. This rule was first brought into consideration by R. Carnap [1]. Carnap's rule uses an infinite set of premises and is therefore inadmissible within the structure of the formal theories of D. Hilbert. The concept of a derivation in a system with the Carnap rule is undecidable. In mathematical logic one uses, for the study of formal arithmetic, the constructive Carnap rule: If there is an algorithm which for a natural number $n$ provides a derivation of the formula $\phi(n)$, then the proposition $\forall x\phi(x)$ can be regarded as being proved (the restricted $\omega$-rule, the rule of constructive infinite induction). Classical arithmetic calculus, which by Gödel's theorem is incomplete, becomes complete on adding the constructive Carnap rule (see [2], [3]).
References
| [1] | R. Carnap, "The logical syntax of language" , Kegan Paul, Trench & Truber (1937) (Translated from German) |
| [2] | A.V. Kuznetsov, Uspekhi Mat. Nauk , 12 : 4 (1957) pp. 218–219 |
| [3] | J.R. Shoenfield, "On a restricted $\omega$-rule" Bull. Acad. Polon. Sci. Cl. III , 7 (1959) pp. 405–407 |