Locally Compact Topological Group에서의 測度에 관하여 = Measures on Locally Compact Topological Group

다국어 초록 (Multilingual Abstract)

If G is a locally compact group and H a locally compact subgroup then H has a Haar measure and operating that measure on H to all the left cosets of H in G give a measure on G, which is a refinement of the Haar measure of G. But we asked that given a measure on G which is a refinement of the Haar measure of G, how can it be get a locally compact subgroup, which is explained as a reverse problem of former. Weil(1947) has proved the reverse problem for measures satisfying Weil's condition, by showing that such a refinement of the Haar measure always give rise to a locally compact subgroup. Of course, the subgroup we obtain need not be unique. But locally compact topology on G which is a refinement of the given topology is unique. In this paper we give a relation between refinement of the measure and given locally compact topology by using the Weil's condition. And we are going to use this topology to characterize another topology. Moreover, we give a difference between regularity of Haar measure and general measure, and give counterexamples about regularity in Haar measure. This problems are;
(1) Can Haar measure be simultaneously inner and outer regular?
(2) Can the product of two regular Borel measure be a regular Borel measure?