On Borel structures in function spaces
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1 On Borel structures in function spaces Witold Marciszewski, Grzegorz Plebanek, and Roman Pol University of Warsaw Interactions between Logic, Topological structures and Banach spaces theory Eilat, May 19 24, 2013 Marciszewski, Plebanek & Pol (UW) Eilat / 14
2 For a compact space K, C(K ) is the Banach space of real-valued continuous functions on K (with the sup norm). Marciszewski, Plebanek & Pol (UW) Eilat / 14
3 For a compact space K, C(K ) is the Banach space of real-valued continuous functions on K (with the sup norm). We will consider the following three σ-algebras in C(K ): Borel(C(K ), norm), Borel(C(K ), weak), Borel(C(K ), pointwise) the σ-algebras of Borel sets in C(K ) with respect to the norm topology, the weak topology, or the pointwise topology in C(K ), respectively. Marciszewski, Plebanek & Pol (UW) Eilat / 14
4 The relations between Borel structures in (C(K ), norm), (C(K ), weak) and (C(K ), pointwise) Marciszewski, Plebanek & Pol (UW) Eilat / 14
5 The relations between Borel structures in (C(K ), norm), (C(K ), weak) and (C(K ), pointwise) Remark For a metrizable compact K we have Borel(C(K ), norm) = Borel(C(K ), weak) = Borel(C(K ), pointwise) Marciszewski, Plebanek & Pol (UW) Eilat / 14
6 The relations between Borel structures in (C(K ), norm), (C(K ), weak) and (C(K ), pointwise) Remark For a metrizable compact K we have Borel(C(K ), norm) = Borel(C(K ), weak) = Borel(C(K ), pointwise) An equivalent norm on C(K ) is a Kadec (pointwise-kadec) norm if the weak (pointwise) topology coincides with the norm topology on the unit sphere {f C(K ) : f = 1}. Marciszewski, Plebanek & Pol (UW) Eilat / 14
7 The relations between Borel structures in (C(K ), norm), (C(K ), weak) and (C(K ), pointwise) Remark For a metrizable compact K we have Borel(C(K ), norm) = Borel(C(K ), weak) = Borel(C(K ), pointwise) An equivalent norm on C(K ) is a Kadec (pointwise-kadec) norm if the weak (pointwise) topology coincides with the norm topology on the unit sphere {f C(K ) : f = 1}. If C(K ) admits a Kadec (pointwise-kadec) norm then Borel(C(K ), norm) = Borel(C(K ), weak) (Borel(C(K ), norm) = Borel(C(K ), pointwise)). Marciszewski, Plebanek & Pol (UW) Eilat / 14
8 The relations between Borel structures in (C(K ), norm), (C(K ), weak) and (C(K ), pointwise) Remark For a metrizable compact K we have Borel(C(K ), norm) = Borel(C(K ), weak) = Borel(C(K ), pointwise) An equivalent norm on C(K ) is a Kadec (pointwise-kadec) norm if the weak (pointwise) topology coincides with the norm topology on the unit sphere {f C(K ) : f = 1}. If C(K ) admits a Kadec (pointwise-kadec) norm then Borel(C(K ), norm) = Borel(C(K ), weak) (Borel(C(K ), norm) = Borel(C(K ), pointwise)). C(K ) admits a pointwise-kadec norm if K belongs to one of the following classes of compacta: Eberlein, Corson, Valdivia, scattered of countable height, linearly ordered, products of linearly ordered. Marciszewski, Plebanek & Pol (UW) Eilat / 14
9 Theorem (Pol and M.) It is consistent with ZFC that there exists a compact space K such that Borel(C(K ), norm) = Borel(C(K ), weak) = Borel(C(K ), pointwise), but C(K ) has no Kadec renorming. Marciszewski, Plebanek & Pol (UW) Eilat / 14
10 Theorem (Pol and M.) It is consistent with ZFC that there exists a compact space K such that Borel(C(K ), norm) = Borel(C(K ), weak) = Borel(C(K ), pointwise), but C(K ) has no Kadec renorming. Question Is it possible to construct such example in ZFC? Marciszewski, Plebanek & Pol (UW) Eilat / 14
11 Theorem (Talagrand) Borel(C(βω), norm) Borel(C(βω), weak). Marciszewski, Plebanek & Pol (UW) Eilat / 14
12 Theorem (Talagrand) Borel(C(βω), norm) Borel(C(βω), weak). Example (Haydon, D. Burke and Pol) There exists a scattered compact space K such that Borel(C(K ), norm) Borel(C(K ), weak) = Borel(C(K ), pointwise). Marciszewski, Plebanek & Pol (UW) Eilat / 14
13 Theorem (Talagrand) Borel(C(βω), norm) Borel(C(βω), weak). Example (Haydon, D. Burke and Pol) There exists a scattered compact space K such that Borel(C(K ), norm) Borel(C(K ), weak) = Borel(C(K ), pointwise). A compact space K is Rosenthal compact if K embeds into B 1 (ω ω ) = {f R ωω : f is of the first Baire class} R ωω. Marciszewski, Plebanek & Pol (UW) Eilat / 14
14 Theorem (Talagrand) Borel(C(βω), norm) Borel(C(βω), weak). Example (Haydon, D. Burke and Pol) There exists a scattered compact space K such that Borel(C(K ), norm) Borel(C(K ), weak) = Borel(C(K ), pointwise). A compact space K is Rosenthal compact if K embeds into B 1 (ω ω ) = {f R ωω : f is of the first Baire class} R ωω. Example (Todorcevic) There exists a scattered Rosenthal compactum T such that Borel(C(T ), norm) Borel(C(T ), weak). Marciszewski, Plebanek & Pol (UW) Eilat / 14
15 Problem Do the Borel structures in C(K ) coincide for separable Rosenthal compacta? Marciszewski, Plebanek & Pol (UW) Eilat / 14
16 Problem Do the Borel structures in C(K ) coincide for separable Rosenthal compacta? This is true if, in addition, K B 1 (ω ω ) consists of functions with at most countably many discontinuity points - in this case C(K ) admits a pointwise-kadec norm (Haydon, Moltó, and Orihuela). Marciszewski, Plebanek & Pol (UW) Eilat / 14
17 Example (Pol and M.) There exists a compact space K such that Borel(C(K ), norm) Borel(C(K ), weak) Borel(C(K ), pointwise). Marciszewski, Plebanek & Pol (UW) Eilat / 14
18 Example (Pol and M.) There exists a compact space K such that Borel(C(K ), norm) Borel(C(K ), weak) Borel(C(K ), pointwise). K is the Stone space of the measure algebra associated with the Lebesgue measure λ on [0, 1] (C(K ) is a function space representation of the algebra L [0, 1]). Marciszewski, Plebanek & Pol (UW) Eilat / 14
19 Example (Pol and M.) There exists a compact space K such that Borel(C(K ), norm) Borel(C(K ), weak) Borel(C(K ), pointwise). K is the Stone space of the measure algebra associated with the Lebesgue measure λ on [0, 1] (C(K ) is a function space representation of the algebra L [0, 1]). If ϕ : K L is a continuous surjection, then the map f f ϕ defines an embedding of C(L) into C(K ) with respect to the norm, weak, and pointwise topologies. Marciszewski, Plebanek & Pol (UW) Eilat / 14
20 Example (Pol and M.) There exists a compact space K such that Borel(C(K ), norm) Borel(C(K ), weak) Borel(C(K ), pointwise). K is the Stone space of the measure algebra associated with the Lebesgue measure λ on [0, 1] (C(K ) is a function space representation of the algebra L [0, 1]). If ϕ : K L is a continuous surjection, then the map f f ϕ defines an embedding of C(L) into C(K ) with respect to the norm, weak, and pointwise topologies. ω = βω \ ω Corollary (CH) Borel(C(ω ), weak) Borel(C(ω ), pointwise). Marciszewski, Plebanek & Pol (UW) Eilat / 14
21 Theorem (Plebanek and M.) Borel(C(βω), weak) Borel(C(βω), pointwise). Marciszewski, Plebanek & Pol (UW) Eilat / 14
22 Theorem (Plebanek and M.) Borel(C(βω), weak) Borel(C(βω), pointwise). Corollary Borel(C(ω ), norm) Borel(C(ω ), weak) Borel(C(ω ), pointwise). Marciszewski, Plebanek & Pol (UW) Eilat / 14
23 Theorem (Plebanek and M.) Borel(C(βω), weak) Borel(C(βω), pointwise). Corollary Borel(C(ω ), norm) Borel(C(ω ), weak) Borel(C(ω ), pointwise). (C(βω), pointwise) has a sequence of open sets separating points. Marciszewski, Plebanek & Pol (UW) Eilat / 14
24 Theorem (Plebanek and M.) Borel(C(βω), weak) Borel(C(βω), pointwise). Corollary Borel(C(ω ), norm) Borel(C(ω ), weak) Borel(C(ω ), pointwise). (C(βω), pointwise) has a sequence of open sets separating points. Theorem (Pol and M.) Let C(K ) be a function space representation of the algebra L [0, 1]. Then no sequence of Borel sets in (C(K ), pointwise) separates points of C(K ). Marciszewski, Plebanek & Pol (UW) Eilat / 14
25 Theorem (Plebanek and M.) Borel(C(βω), weak) Borel(C(βω), pointwise). Corollary Borel(C(ω ), norm) Borel(C(ω ), weak) Borel(C(ω ), pointwise). (C(βω), pointwise) has a sequence of open sets separating points. Theorem (Pol and M.) Let C(K ) be a function space representation of the algebra L [0, 1]. Then no sequence of Borel sets in (C(K ), pointwise) separates points of C(K ). Corollary The Banach spaces C(K ) and C(βω) are isomorphic, but there is no Borel-measurable injection ϕ : (C(K ), pointwise) (C(βω), pointwise). Marciszewski, Plebanek & Pol (UW) Eilat / 14
26 Theorem (Plebanek and M.) No sequence of Borel sets in (C(ω ), pointwise) separates points of C(ω ). Marciszewski, Plebanek & Pol (UW) Eilat / 14
27 Theorem (Plebanek and M.) No sequence of Borel sets in (C(ω ), pointwise) separates points of C(ω ). Corollary There is no Borel-measurable injection ϕ : (C(ω ), pointwise) (C(βω), pointwise). Marciszewski, Plebanek & Pol (UW) Eilat / 14
28 Problem Let Borel(C(K ), norm) = Borel(C(K ), weak). Is it true that also Borel(C(K ), weak) = Borel(C(K ), pointwise)? Marciszewski, Plebanek & Pol (UW) Eilat / 14
29 Problem Let Borel(C(K ), norm) = Borel(C(K ), weak). Is it true that also Borel(C(K ), weak) = Borel(C(K ), pointwise)? Problem Let Borel(C(K ), norm) Borel(C(K ), weak). Does there exist a norm-discrete set in C(K ) which is not Borel with respect to the weak topology in C(K )? Marciszewski, Plebanek & Pol (UW) Eilat / 14
30 Problem Let Borel(C(K ), norm) = Borel(C(K ), weak). Is it true that also Borel(C(K ), weak) = Borel(C(K ), pointwise)? Problem Let Borel(C(K ), norm) Borel(C(K ), weak). Does there exist a norm-discrete set in C(K ) which is not Borel with respect to the weak topology in C(K )? This is true if K is scattered (D. Burke and R. Pol). Marciszewski, Plebanek & Pol (UW) Eilat / 14
31 A nonnegative Radon measure µ on a compact space K is a category measure, if µ-null sets coincide with meager sets in K. Marciszewski, Plebanek & Pol (UW) Eilat / 14
32 A nonnegative Radon measure µ on a compact space K is a category measure, if µ-null sets coincide with meager sets in K. If ν is a non-atomic probability measure and C(K ) is a function space representation of the algebra L (ν) (K is the Stone space of the measure algebra associated with ν), then K is an extremally disconnected compact space and ν gives rise to a non-atomic probability category measure on K. Marciszewski, Plebanek & Pol (UW) Eilat / 14
33 A nonnegative Radon measure µ on a compact space K is a category measure, if µ-null sets coincide with meager sets in K. If ν is a non-atomic probability measure and C(K ) is a function space representation of the algebra L (ν) (K is the Stone space of the measure algebra associated with ν), then K is an extremally disconnected compact space and ν gives rise to a non-atomic probability category measure on K. Theorem Let µ be a non-atomic probability category measure on an extremally disconnected compact space K. Then the set {f C(K ) : fdµ > 0} is not a Borel set in the pointwise topology on C(K ). Marciszewski, Plebanek & Pol (UW) Eilat / 14
34 Given a bounded sequence (x n ) n ω and an ultrafilter ω, by lim x n we denote the -limit of (x n ). Marciszewski, Plebanek & Pol (UW) Eilat / 14
35 Given a bounded sequence (x n ) n ω and an ultrafilter ω, by lim x n we denote the -limit of (x n ).For any ultrafilter ω, we define the finitely additive measure d on ω by the formula for A ω. d (A) = lim A n n, Marciszewski, Plebanek & Pol (UW) Eilat / 14
36 Given a bounded sequence (x n ) n ω and an ultrafilter ω, by lim x n we denote the -limit of (x n ).For any ultrafilter ω, we define the finitely additive measure d on ω by the formula for A ω. d (A) = lim A n n Let d be a (uniquely determined) Radon measure on βω such that d (A) = d (A), for any A P(ω), where A is the closure of A in βω., Marciszewski, Plebanek & Pol (UW) Eilat / 14
37 Given a bounded sequence (x n ) n ω and an ultrafilter ω, by lim x n we denote the -limit of (x n ).For any ultrafilter ω, we define the finitely additive measure d on ω by the formula for A ω. d (A) = lim A n n Let d be a (uniquely determined) Radon measure on βω such that d (A) = d (A), for any A P(ω), where A is the closure of A in βω. Theorem (Plebanek, M.) For any ultrafilter ω, the set {f C(βω) : fd d > 0} is not a Borel set in the pointwise topology on C(βω)., Marciszewski, Plebanek & Pol (UW) Eilat / 14
38 σ-algebras of Baire sets in function spaces Marciszewski, Plebanek & Pol (UW) Eilat / 14
39 σ-algebras of Baire sets in function spaces For a topological space X the Baire σ-algebra Ba(X) is generated by all continuous real-valued functions on X. Marciszewski, Plebanek & Pol (UW) Eilat / 14
40 σ-algebras of Baire sets in function spaces For a topological space X the Baire σ-algebra Ba(X) is generated by all continuous real-valued functions on X. For a compact space K, we denote by Ba(C(K ), weak), Ba(C(K ), pointwise) the Baire σ-algebras in C(K ) endowed with the weak topology, or the pointwise topology, respectively. Marciszewski, Plebanek & Pol (UW) Eilat / 14
41 σ-algebras of Baire sets in function spaces For a topological space X the Baire σ-algebra Ba(X) is generated by all continuous real-valued functions on X. For a compact space K, we denote by Ba(C(K ), weak), Ba(C(K ), pointwise) the Baire σ-algebras in C(K ) endowed with the weak topology, or the pointwise topology, respectively. Ba(C(K ), pointwise) Ba(C(K ), weak) Borel(C(K ), pointwise) Borel(C(K ), weak) Borel(C(K ), norm) Marciszewski, Plebanek & Pol (UW) Eilat / 14
42 σ-algebras of Baire sets in function spaces For a topological space X the Baire σ-algebra Ba(X) is generated by all continuous real-valued functions on X. For a compact space K, we denote by Ba(C(K ), weak), Ba(C(K ), pointwise) the Baire σ-algebras in C(K ) endowed with the weak topology, or the pointwise topology, respectively. Ba(C(K ), pointwise) Ba(C(K ), weak) Borel(C(K ), pointwise) Borel(C(K ), weak) Borel(C(K ), norm) Proposition (Plebanek and M.) There exists a pointwise open subset of C(βω) which does not belong to Ba(C(βω), weak). Marciszewski, Plebanek & Pol (UW) Eilat / 14
43 Ba(C(βω), pointwise) Ba(C(βω), weak) Borel(C(βω), pointwise) Borel(C(βω), weak) Borel(C(βω), norm) Marciszewski, Plebanek & Pol (UW) Eilat / 14
44 Ba(C(βω), pointwise) Ba(C(βω), weak) Borel(C(βω), pointwise) Borel(C(βω), weak) Borel(C(βω), norm) Also for K = ω or for the Stone space of the measure algebra associated with the Lebesgue measure on [0, 1], all five σ-algebras in C(K ) from the above diagram are distinct. Marciszewski, Plebanek & Pol (UW) Eilat / 14
45 Ba(C(βω), pointwise) Ba(C(βω), weak) Borel(C(βω), pointwise) Borel(C(βω), weak) Borel(C(βω), norm) Also for K = ω or for the Stone space of the measure algebra associated with the Lebesgue measure on [0, 1], all five σ-algebras in C(K ) from the above diagram are distinct. Theorem (Aviles, Plebanek, Rodriguez) Ba(C(2 ω 1 ), pointwise) = Borel(C(2 ω 1 ), norm). Marciszewski, Plebanek & Pol (UW) Eilat / 14
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