The existence of invariant measures on certain quotient spaces

On the Existence of Invariant Measures

Let P b e a measurable transformation on a measure space (Q, Cfc, P ) , with 0<P(£2) < oo. Call T absolutely continuous if P(A) = 0 implies P(T~*A)=0. The transformation T is said to have the Birkhoff recurrence property if, for each -4EG*, limn-co(lA0 2"~o XA{T 'CO) exists for almost all co£0. I t has been shown that if T is absolutely continuous and has the Birkhoff recurrence property, then ...

‎On the two-wavelet localization operators on homogeneous spaces with relatively invariant measures

In ‎the present ‎paper, ‎we ‎introduce the ‎two-wavelet ‎localization ‎operator ‎for ‎the square ‎integrable ‎representation ‎of a‎ ‎homogeneous space‎ with respect to a relatively invariant measure. ‎We show that it is a bounded linear operator. We investigate ‎some ‎properties ‎of the ‎two-wavelet ‎localization ‎operator ‎and ‎show ‎that ‎it ‎is a‎ ‎compact ‎operator ‎and is ‎contained ‎in‎ a...

On Certain Conditions for the Existence of Invariant Linear Functionals

Invariant Chains and the Homology of Quotient Spaces

For a finite group G and a finite G-CW-complex X, we construct groups H•(G, X) as the homology groups of the G-invariants of the cellular chain complex C•(X). These groups are related to the homology of the quotient space X/G via a norm map, and therefore provide a mechanism for calculating H•(X/G). We compute several examples and provide a new proof of “Smith theory”: if G = Z/p and X is a mod...

The Existence of Universal Invariant Semiregular Measures on Groups

A nonnegative, countably additive, extended real-valued measure is universal on a set X iff it is defined on all subsets of X, and is semiregular iff every set of positive measure contains a subset of positive finite measure. We prove that on every group of sufficiently large cardinality there exists a universal invariant semiregular measure vanishing on singletons. Thus we give complete soluti...