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Permit okay = ∪i∈I Ki be any σ-finite set, the place every one Ki is of finite degree. we have to exhibit that μM constrained to the subalgebra ΣK = {E ∩ ok : E ∈ Σ} is only atomic. feel to the contrary that it's not. Then there's an atomless set N of confident degree in ΣK . For a few i0 ∈ I the intersection N ∩ Ki0 has optimistic degree. truly, N0 := N ∩ Ki0 can be atomless, as a result we will break up this set into units of optimistic degree. The smallest in degree of those units, say N1 , is of degree μM (N1 ) ≤ μM (N0 )/2. carrying on with this manner we receive units of arbitrarily small degree, contradicting (3. 4). ✷ From (3. four) we see that for norm-bounded Riesz households, the atoms Mi within the illustration (3. three) are bounded from less than in degree. for this reason, no matter if we examine the sum in (3. three) as a Riemann variety sum, there's a guaranteed to any refinement. If we imagine that M is a Hausdorff topological team and that μM satisfies convinced vulnerable regularity assumptions, e. g. , M being a in the community compact crew with the standard left Haar degree, then the life of a norm-bounded Riesz kinfolk forces the gang M to be discrete. Proposition three. four. enable M be a Hausdorff topological crew with a left Haar degree μM (as defined by means of Fremlin [17, Def. 441D]). If {fk }k∈M is a norm bounded uncomplicated Riesz kin, then M is a discrete crew. evidence. From Proposition 443O in [17] we all know that μM isn't really non-atomic if and provided that there's the discrete topology on M . in spite of the fact that, by means of Proposition three. three, the degree μM is obviously no longer non-atomic, therefore the end result follows. ✷ 240 M. S. Jakobsen, J. Lemvig / magazine of practical research 270 (2016) 229–263 We finish this part with a Riesz relations variation of Theorem three. 2. Theorem three. five. (See [28]. ) enable H be a Hilbert house, permit A, B > zero, and permit M be a countable index set outfitted with the counting degree. Then the subsequent statements are similar: (i) {fk }k∈M is a easy Riesz relatives (i. e. , a Riesz series) in H with bounds A and B; (ii) {fk }k∈M is a Bessel relations in H with sure B and there exists a Bessel relatives {gk }k∈M in H with certain A−1 such that fk , g = δk, , ok, ∈ M . four. Gabor structures The Gabor approach G (g, Δ) = {π(ν)g}ν∈Δ is general whilst Δ is a closed subgroup of G × G. If Δ isn't a subgroup, e. g. , only a collection of issues, the Gabor method is abnormal. If Δ = Λ × Γ for closed subgroups Λ ⊂ G and Γ ⊂ G, we are saying that G (g, Δ) = {Eγ Tλ g}λ∈Λ,γ∈Γ is a separable Gabor approach. If Δ isn't assumed to have this kind, G (g, Δ) is non-separable. during this paintings we will reflect on non-separable, general Gabor structures. The research, synthesis, and the (mixed) body operator for Gabor Bessel structures are defined as in part three. specifically, the (mixed) body operator for 2 Gabor Bessel platforms generated through the capabilities g, h ∈ L2 (G) takes the shape Sg,h : L2 (G) → L2 (G), · , π(ν)g π(ν)h dν. Sg,h = Δ If g = h, we get well the body operator Sg = Sg,g , additionally easily denoted through S. it really is simple to teach that the body operator commutes with time–frequency shifts with recognize to the crowd Δ. Lemma four.