Harmonic Analysis - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-18T22:04:24Z http://mathoverflow.net/feeds/question/106515 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106515/harmonic-analysis Harmonic Analysis esmaeelzadeh 2012-09-06T14:42:30Z 2012-09-06T17:11:32Z <p>Let ‎$‎G‎$ ‎be a‎ ‎locally ‎compact ‎group‎, ‎$‎H‎$ ‎be a‎ ‎closed ‎subgroup ‎and ‎‎$‎N‎$ ‎be a‎ ‎normal ‎subgroup ‎of ‎‎$‎G‎$ ‎such ‎that ‎‎$‎H‎\subseteq ‎N‎$‎. ‎How ‎can ‎we get $$\int_{G/H} ‎f(xH)d‎\mu_{G/H}‎(xH)=‎\mu_{N/H}‎(N/H)‎‎‎‎‎‎‎\int_{G/N}f(xN)d‎\mu_{G/N}(xN)\ \ ?$$</p> http://mathoverflow.net/questions/106515/harmonic-analysis/106531#106531 Answer by anton for Harmonic Analysis anton 2012-09-06T17:11:32Z 2012-09-06T17:11:32Z <p>First, in order for an invariant measure on $G/H$ to exist, you need for the modular functions the equality $$\Delta_G|_H=\Delta_H.$$ Under these circumstances the left hand side if the equation is defined, provided the function $f$ is $H$-invariant. If in addition $f$ is even $N$-invariant the right hand side is defined as well (the modular condition is automatic, as $N$ is normal). If the measure $\mu_{N/H}(N/H)$ is finite, it then is possible to choose the invariant measures such that the claimed equality holds for every measurable $N$-invariant function which is such that either side of the equality exists. A proof can be found in Deitmar/Echterhoff: Principles of Harmonic Analysis.</p>