On extension of Baire submeasures.

*(English)*Zbl 0606.28010Let T be a locally compact topological space. It is well known that each Baire measure on it can be extended uniquely to a regular Borel measure. In a previous paper of the author [Diss. Math. 112, 35 p. (1974; Zbl 0292.28001)] this result was generalized to submeasures. In the present paper a new proof is presented. It is based on the following observation. If \(\mu\) is a regular Borel (sub)measure, then to each Borel set A there is a Baire set E such that \(A\Delta E\) is a \(\mu\)-null set. Thus to extend a Baire submeasure it suffices to define appropriate \(\sigma\)- ideal of ”null”sets. This method is known at least since P. A. Meyer’s ”Probabilités et potentiel” (1966; Zbl 0138.104)]. The definition of submeasure used here does not contain subadditivity, but by the work of L. Drewnowski [Colloq. Math. 38, 243-253 (1978; Zbl 0398.28003)] this is in fact inessential. The proofs are in the same vein as in P. R. Halmos’ ”Measure theory” (1950; Zbl 0040.168)].

Reviewer: B.Aniszczyk

##### MSC:

28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |

28A12 | Contents, measures, outer measures, capacities |

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##### References:

[1] | BERBERIAN S. K.: Measure and Integration. New York-London 1965. · Zbl 0126.08001 |

[2] | BERBERIAN S. K.: On the extension of Borel Measures. Proc. Amer. Math. Soc. 16 1965, 415-418. · Zbl 0145.28101 |

[3] | DOBRAKOV I.: On submeasures I. Dissertationes Math. 112, Warszawa 1974. · Zbl 0292.28001 |

[4] | DOBRAKOV I., FARKOVÁ J.: On submeasures II. Math. Slovaca 30 1980, 65-81. · Zbl 0428.28001 |

[5] | DOBRAKOV I.: A concept of measurability for the Daniell integral. Math. Slovaca 28 1978, 361-378. · Zbl 0414.28009 |

[6] | HALMOS P. R.: Measure Theory. New York 1950. · Zbl 0040.16802 |

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