Let F be a locally compact group and K a Banach space. The left L1^) module K is by definition absolutely continuous under the composition * if for k e K there exist f e LX{T), k' e Kviith k =f * k'. If the locally compact Hausdorff space X is a transformation group over Y and has a measure quasi-invariant with respect to T, then L*(X) is an absolutely continuous L1(r) module-the main object we study. If Y<£X is measurable, let LY consist of all functions in L1(X) vanishing outside Y. For OsT

more »

... t locally null and B a closed linear subs pace of K, we observe the connection between the closed linear span (denoted La * B) of the elements/* k, with fe La and k e B, and the collection of functions of B shifted by elements in Ci. As a result, a closed linear subspace of L1(Y) is an Lz for some measurable Z<=X if and only if it is closed under pointwise multiplication by elements of L"(X). This allows the theorem stating that if Clç^T and KÇJTare both measurable, then there is a measurable subset Z of X such that La * LY = LZ. Under certain restrictions on T, we show that this Z is essentially open in the (usually stronger) orbit topology on X. Finally we prove that if Ci and Y are both relatively sigma-compact, and if also La * Ly^Ly, then there exist Hi and Yx locally almost everywhere equal to Ci and Y respectively, such that Cl¡ Yx £ Yx ; in addition we characterize those Cl