{"title":"无穷 nary 群的哈量","authors":"M. Shahryari, M. Rostami","doi":"10.1142/s0219498825502196","DOIUrl":null,"url":null,"abstract":"<p>We prove that every profinite <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>-ary group <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>G</mi><mo>,</mo><mi>f</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mrow><mstyle><mtext mathvariant=\"normal\">der</mtext></mstyle></mrow><mrow><mi>𝜃</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo>,</mo><mo stretchy=\"false\">•</mo><mo stretchy=\"false\">)</mo></math></span><span></span> has a unique Haar measure <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span> and further for every measurable subset <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi><mo>⊆</mo><mi>G</mi></math></span><span></span>, we have <disp-formula-group><span><math altimg=\"eq-00007.gif\" display=\"block\" overflow=\"scroll\"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>m</mi><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">−</mo><mn>1</mn><mo stretchy=\"false\">)</mo><msup><mrow><mi>m</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo><mo>,</mo></mrow></math></span><span></span></disp-formula-group> where <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi></math></span><span></span> and <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>m</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup></math></span><span></span> are the normalized Haar measures of the profinite groups <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>G</mi><mo>,</mo><mo stretchy=\"false\">•</mo><mo stretchy=\"false\">)</mo></math></span><span></span> and the Post cover <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>G</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup></math></span><span></span>, respectively.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Haar measure of a profinite n-ary group\",\"authors\":\"M. Shahryari, M. Rostami\",\"doi\":\"10.1142/s0219498825502196\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that every profinite <span><math altimg=\\\"eq-00003.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>n</mi></math></span><span></span>-ary group <span><math altimg=\\\"eq-00004.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo stretchy=\\\"false\\\">(</mo><mi>G</mi><mo>,</mo><mi>f</mi><mo stretchy=\\\"false\\\">)</mo><mo>=</mo><msub><mrow><mstyle><mtext mathvariant=\\\"normal\\\">der</mtext></mstyle></mrow><mrow><mi>𝜃</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy=\\\"false\\\">(</mo><mi>G</mi><mo>,</mo><mo stretchy=\\\"false\\\">•</mo><mo stretchy=\\\"false\\\">)</mo></math></span><span></span> has a unique Haar measure <span><math altimg=\\\"eq-00005.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span> and further for every measurable subset <span><math altimg=\\\"eq-00006.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>A</mi><mo>⊆</mo><mi>G</mi></math></span><span></span>, we have <disp-formula-group><span><math altimg=\\\"eq-00007.gif\\\" display=\\\"block\\\" overflow=\\\"scroll\\\"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\\\"false\\\">(</mo><mi>A</mi><mo stretchy=\\\"false\\\">)</mo><mo>=</mo><mi>m</mi><mo stretchy=\\\"false\\\">(</mo><mi>A</mi><mo stretchy=\\\"false\\\">)</mo><mo>=</mo><mo stretchy=\\\"false\\\">(</mo><mi>n</mi><mo stretchy=\\\"false\\\">−</mo><mn>1</mn><mo stretchy=\\\"false\\\">)</mo><msup><mrow><mi>m</mi></mrow><mrow><mo stretchy=\\\"false\\\">∗</mo></mrow></msup><mo stretchy=\\\"false\\\">(</mo><mi>A</mi><mo stretchy=\\\"false\\\">)</mo><mo>,</mo></mrow></math></span><span></span></disp-formula-group> where <span><math altimg=\\\"eq-00008.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>m</mi></math></span><span></span> and <span><math altimg=\\\"eq-00009.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><msup><mrow><mi>m</mi></mrow><mrow><mo stretchy=\\\"false\\\">∗</mo></mrow></msup></math></span><span></span> are the normalized Haar measures of the profinite groups <span><math altimg=\\\"eq-00010.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mo stretchy=\\\"false\\\">(</mo><mi>G</mi><mo>,</mo><mo stretchy=\\\"false\\\">•</mo><mo stretchy=\\\"false\\\">)</mo></math></span><span></span> and the Post cover <span><math altimg=\\\"eq-00011.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><msup><mrow><mi>G</mi></mrow><mrow><mo stretchy=\\\"false\\\">∗</mo></mrow></msup></math></span><span></span>, respectively.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-03-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219498825502196\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825502196","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We prove that every profinite -ary group has a unique Haar measure and further for every measurable subset , we have where and are the normalized Haar measures of the profinite groups and the Post cover , respectively.