Weil-Nachbin Theory for Locally Compact Groups

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Abstract: Assume $$H$$ is a normal subgroup of the locally compact group $$G$$ with measures $$μ$$ and $$η$$ the on $$G$$ and $$H$$, respectively, and assume $$m:G\longrightarrow R_+$$ is a continuous homomorphism from $$G$$ to a group $$R_+$$ of positive real numbers with the operation of multiplication, then we establish that for the existence of a measure $$\partial$$ on the quotient group $$G/H$$, it is necessary and sufficient that $$Δ^{r}_H(h)=m(h)Δ^{r}_G(h)$$ holds for all $$h \in H $$ so that each -relatively invariant measure on $$G/H$$ is a quotient measure $$μ_η=μ/η$$; also, we show that the -relatively invariant measure $$\partial$$ on $$G/H$$ can be presented in the form $$\partial=ρ(mμ)$$ where $$ρ:G\longrightarrow G/H$$ is a projection mapping.

Proof, ISSN / E-ISSN: 2944-9162 / 2732-9941, Volume 2, 2022, Art. #23

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Mykola Ivanovich Yaremenko, "Weil-Nachbin Theory for Locally Compact Groups," Proof, vol. 2, pp. 174-180, 2022, DOI:10.37394/232020.2022.2.23
Mykola Ivanovich Yaremenko. Weil-Nachbin Theory for Locally Compact Groups. Proof. 2022;2:174-180. 10.37394/232020.2022.2.23
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