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Everything the deals with properties and definitions of Haar measure, as well as related fields when the question relies heavily on the notion of haar measure - group harmonic analysis, group ergodic theory etc.
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When is $L^2(X)$ separable?
Without loss of generality we can assume that the support of the measure equals $X$
(i.e., the measure is faithful),
because we can always pass to the subspace defined by the support of the measure.
T …