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Theorem 4.8.2 (Miller[84]) Let \( G \) be a Polish group and \( H \) a Borel subgroup. Suppose the \( \sigma \) -algebra of invariant Borel sets is countably generated. Then \( H \) is closed. | Proof of 4.8.2. Let \( X = G/H \), the set of right cosets, and \( q : G \rightarrow \) \( G/H \) the quotient map. Equip \( G/H \) with the largest \( \sigma \) -algebra making \( q \) Borel measurable. By our hypothesis, \( X \) is a countably generated measurable space with singletons as atoms. Consider the action \... | Yes |
Proposition 4.8.3 Let \( X \) be a Polish space and \( G \) a group of homeomorphisms of \( X \) such that for every pair \( U, V \) of nonempty open sets there is a \( g \in G \) with \( g\left( U\right) \cap V \neq \varnothing \) . Suppose \( A \) is a \( G \) -invariant Borel set; i.e., \( g\left( A\right) = A \) fo... | Proof. Suppose neither \( A \) nor \( {A}^{c} \) is meager in \( X \) . Then there exist nonempty open sets \( U, V \) such that \( A \) and \( {A}^{c} \) are comeager in \( U \) and \( V \) respectively. By our hypothesis, there is a \( g \in G \) such that \( g\left( U\right) \cap V \neq \varnothing \) . Let \( W = g... | Yes |
Theorem 4.8.4 (Miller[84]) Let \( \\left( {G, \\cdot }\\right) \) be a Polish group, \( X \) a second countable \( {T}_{1} \) space, and \( \\left( {g, x}\\right) \\rightarrow g \\cdot x \) an action of \( G \) on \( X \) . Suppose that for a given \( x \), the map \( g \\rightarrow g \\cdot x \) is Borel. Then the sta... | Proof. Let \( H = \\operatorname{cl}\\left( {G}_{x}\\right) \) . It is fairly easy to see that we can replace \( G \) by \( H \) . Hence, without loss of generality we assume that \( {G}_{x} \) is dense in \( G \) . \n\nSince \( X \) is second countable and \( {T}_{1},{G}_{x} \) is Borel. Therefore, by 3.5.13, we shall... | Yes |
Theorem 4.8.6 Let \( G \) be a Polish group, \( X \) a Polish space, and \( a\left( {g, x}\right) = \) \( g \cdot x \) an action of \( G \) on \( X \) . Assume that \( g \cdot x \) is continuous in \( x \) for all \( g \) and Borel in \( g \) for all \( x \) . Then the action is continuous. | Proof. By 3.1.30, the action \( a : G \times X \rightarrow X \) is Borel. Let \( \left( {V}_{n}\right) \) be a countable base for \( X \) . Put \( {C}_{n} = {a}^{-1}\left( {V}_{n}\right) \) . Then \( {C}_{n} \) is Borel with open sections. By 4.7.2, write\n\n\[ \n{C}_{n} = \mathop{\bigcup }\limits_{m}\left( {{B}_{nm} \... | Yes |
Lemma 4.8.8 If \( \left( {G, \cdot }\right) \) is a group with a Polish topology such that the group operation \( \left( {g, h}\right) \rightarrow g \cdot h \) is Borel, then \( g \rightarrow {g}^{-1} \) is continuous. | Proof. Since \( \left( {g, h}\right) \rightarrow g \cdot h \) is Borel, the graph\n\n\[ \n\{ \left( {g, h}\right) : g \cdot h = e\}\n\]\n\nof \( g \rightarrow {g}^{-1} \) is Borel. Hence, by 4.5.2, \( g \rightarrow {g}^{-1} \) is Borel measurable. An imitation of the proof of 3.5.9 shows that \( g \rightarrow {g}^{-1} ... | No |
Proposition 4.8.9 If \( \left( {G, \cdot }\right) \) is a group with a Polish topology such that the group operation is separately continuous in each variable, then \( G \) is a topological group. | Proof. In view of 4.8.8, we have only to show that the group operation is jointly continuous. This we get immediately by applying 4.8.6 to \( X = G \) and action \( g \cdot x \) the group operation. | Yes |
Theorem 4.8.10 (S. Solecki and S. M. Srivastava[109]) Let \( \left( {G, \cdot }\right) \) be a group with a Polish topology such that \( h \rightarrow g \cdot h \) is continuous for every \( g \in G \), and \( g \rightarrow g \cdot h \) Borel for all \( h \) . Then \( G \) is a topological group. | Proof. By 4.8.9, we only have to show that the group operation \( g \cdot h \) is jointly continuous. A close examination of the proof of 4.8.6 shows that this follows from the following result.\n\nLemma 4.8.11 Let \( G \) satisfy the hypothesis of our theorem. Then for every meager set \( I \) and every \( g \) ,\n\n\... | Yes |
Lemma 4.8.11 Let \( G \) satisfy the hypothesis of our theorem. Then for every meager set \( I \) and every \( g \) , \[ {Ig} = \{ h \cdot g : h \in I\} \] is meager. | Proof. Claim. If \( I \) is meager in \( G \), so is \( {I}^{-1} = \left\{ {h \in G : {h}^{-1} \in I}\right\} \) . Assuming the claim, we prove the lemma as follows. Let \( I \) be meager in \( G \) and \( g \in G \) . By the claim, \( {I}^{-1} \) is meager. Since the group operation is continuous in the second varible... | No |
Lemma 4.9.3 Let \( X \) be a Polish space, \( A \subseteq X \) coanalytic, and \( \varphi \) a norm on \( A \) . Then \( \varphi \) is a \( {\mathbf{\Pi }}_{1}^{1} \) -norm if and only if both \( { \leq }_{\varphi }^{ * },{ < }_{\varphi }^{ * } \) are coanalytic. | Proof. We first prove the \ | No |
Example 4.9.4 Let \( X = {2}^{\mathbb{N} \times \mathbb{N}} \) and \( A = {WO} \). For \( x \in {WO} \), Let \( \left| x\right| < {\omega }_{1} \) be the order type of \( x \). | For \( x \in {2}^{\mathbb{N} \times \mathbb{N}} \), define\n\n\[ m{ < }_{x}n \Leftrightarrow x\left( {m, n}\right) = 1\& x\left( {n, m}\right) = 0. \]\n\nFor \( x, y \) in \( {2}^{\mathbb{N} \times \mathbb{N}} \), set\n\n\[ x{ \leq }_{\left| \cdot \right| }^{{\sum }_{1}^{1}}y \Leftrightarrow \exists z \in {\mathbb{N}}^... | Yes |
Theorem 4.9.8 (Boundedness theorem for \( {\mathbf{\Pi }}_{1}^{1} \) -norms) Suppose \( A \) is a \( {\mathbf{\Pi }}_{1}^{1} \) set in a Polish space \( X \) and \( \varphi \) a norm on \( A \) as defined in 4.9.1. Then for every \( {\mathbf{\sum }}_{1}^{1} \) set \( B \subseteq A,\sup \{ \varphi \left( x\right) : x \i... | Proof. Suppose \( \sup \{ \varphi \left( y\right) : y \in B\} = {\omega }_{1} \) . Take any \( {\mathbf{\Pi }}_{1}^{1} \) set \( C \) that is not \( {\mathbf{\sum }}_{1}^{1} \) . Fix a Borel function \( g \) such that\n\n\[ x \in C \Leftrightarrow g\left( x\right) \in {WO}. \]\n\nThen,\n\n\[ x \in C\; \Leftrightarrow \... | Yes |
Example 4.9.11 (A. Maitra and C. Ryll-Nardzewski[76]) Let \( X, Y \) be uncountable Polish spaces. Let \( U \subseteq X \times X \) be universal analytic and \( C \subseteq Y \) an uncountable coanalytic set not containing a perfect set. We mentioned earlier that Gödel's axiom of constructibility implies the existence ... | Here is a proof. Suppose they are Borel isomorphic. Take a Borel isomorphism \( f : U \rightarrow A \) . By 3.3.5, there exist Borel sets \( {B}_{1} \supseteq U,{B}_{2} \supseteq A \) and a Borel isomorphism \( g : {B}_{1} \rightarrow {B}_{2} \) extending \( f \) . Let \( \varphi \) be a \( {\mathbf{\Pi }}_{1}^{1} \) n... | Yes |
Theorem 4.9.14 (The reduction principle for coanalytic sets) (Kuratowski) Let \( \\left( {A}_{n}\\right) \) be sequence of \( {\\mathbf{\\Pi }}_{1}^{1} \) sets in a Polish space \( X \) . Then there is a sequence \( \\left( {A}_{n}^{ * }\\right) \) of \( {\\mathbf{\\Pi }}_{1}^{1} \) sets such that they are pairwise dis... | Proof. Consider \( A \\subseteq X \\times \\mathbb{N} \) given by\n\n\[ \n\\left( {x, n}\\right) \\in A \\Leftrightarrow x \\in {A}_{n} \n\] \n\nClearly, \( A \) is \( {\\mathbf{\\Pi }}_{1}^{1} \) with projection \( \\mathop{\\bigcup }\\limits_{n}{A}_{n} \) . Let \( \\varphi \) be a \( {\\mathbf{\\Pi }}_{1}^{1} \)-norm... | Yes |
Corollary 4.9.15 Let \( X \) be Polish and \( {A}_{0},{A}_{1} \) coanalytic subsets of \( X \) . Then there exist pairwise disjoint coanalytic sets \( {A}_{0}^{ * },{A}_{1}^{ * } \) contained in \( {A}_{0} \) , \( {A}_{1} \) respectively such that \( {A}_{0}^{ * }\bigcup {A}_{1}^{ * } = {A}_{0}\bigcup {A}_{1} \) . | Proof. In the above theorem, take \( {A}_{n} = \varnothing \) for \( n > 1 \) . | Yes |
Theorem 4.9.19 Let \( X \) be a Polish space. Then there exist sets \( C \in \) \( {\mathbf{\Pi }}_{1}^{1}\left( {\mathbb{N}}^{\mathbb{N}}\right) \) and \( V \in {\mathbf{\Pi }}_{1}^{1}\left( {{\mathbb{N}}^{\mathbb{N}} \times X}\right), U \in {\mathbf{\sum }}_{1}^{1}\left( {{\mathbb{N}}^{\mathbb{N}} \times X}\right) \)... | Proof. Let \( {W}_{0},{W}_{1} \) be coanalytic subsets of \( {\mathbb{N}}^{\mathbb{N}} \times X \) such that for every pair \( \left( {{C}_{0},{C}_{1}}\right) \) of sets in \( {\mathbf{\Pi }}_{1}^{1}\left( X\right) \) there is an \( \alpha \) with \( {C}_{i} = {\left( {W}_{i}\right) }_{\alpha }, i = 0 \) or 1 . By the ... | Yes |
Example 4.10.1 Let \( \mu \) be a finite Borel measure on a Polish space \( X \) and \( {\mu }^{ * } \) the associated outer measure. Thus, for any \( A \subseteq X \) , \[ {\mu }^{ * }\left( A\right) = \inf \{ \mu \left( B\right) : B \supseteq A, B\text{ Borel }\} . \] | It is easy to check that \( {\mu }^{ * } \) is a capacity on \( X \) . | No |
Proposition 4.10.4 Let \( I \) be a capacity on a Polish space \( X \) and that \( {I}^{ * } : \mathcal{P}\left( X\right) \rightarrow \left\lbrack {0,\infty }\right\rbrack \) be defined by\n\n\[ \n{I}^{ * }\left( A\right) = \inf \{ I\left( B\right) : B \supseteq A, B\text{ Borel }\} .\n\]\n\nThen \( {I}^{ * } \) is a c... | Proof. Clearly, \( {I}^{ * } \) is monotone. Further, \( {I}^{ * } \) and \( I \) coincide on Borel sets. As \( I \) is a capacity, it follows that \( {I}^{ * }\left( K\right) < \infty \) for every compact \( K \) and that \( {I}^{ * } \) is right-continuous over compacta.\n\nTo show that \( {I}^{ * } \) is going up, t... | Yes |
Proposition 4.10.4 Let \( I \) be a capacity on a Polish space \( X \) and that \( {I}^{ * } : \mathcal{P}\left( X\right) \rightarrow \left\lbrack {0,\infty }\right\rbrack \) be defined by\n\n\[ \n{I}^{ * }\left( A\right) = \inf \{ I\left( B\right) : B \supseteq A, B\text{ Borel }\} .\n\]\n\nThen \( {I}^{ * } \) is a c... | Proof. Clearly, \( {I}^{ * } \) is monotone. Further, \( {I}^{ * } \) and \( I \) coincide on Borel sets. As \( I \) is a capacity, it follows that \( {I}^{ * }\left( K\right) < \infty \) for every compact \( K \) and that \( {I}^{ * } \) is right-continuous over compacta.\n\nTo show that \( {I}^{ * } \) is going up, t... | Yes |
Proposition 4.10.10 Let \( I \) be a capacity on a Polish space \( X \) and \( A \subseteq X \) universally capacitable. Then\n\n\[ I\left( A\right) = {I}^{ * }\left( A\right) \]\n\nwhere \( {I}^{ * } \) is as defined in 4.10.4. | Proof. By 4.10.4, \( {I}^{ * } \) is a capacity. Now note the following.\n\n\[ {I}^{ * }\left( A\right) = \sup \left\{ {{I}^{ * }\left( K\right) : K \subseteq A\text{ compact }}\right\} \;\text{ (as }A\text{ is }{I}^{ * } - \text{ capacitable) }\n\n= \;\sup \{ I\left( K\right) : K \subseteq A\text{ compact}\} \n\n= I\l... | Yes |
Proposition 4.10.11 \( {\mathbb{N}}^{\mathbb{N}} \) is universally capacitable. | Proof. For any \( s = \left( {{n}_{0},{n}_{1},\ldots ,{n}_{k - 1}}\right) \in {\mathbb{N}}^{ < \mathbb{N}} \), set\n\n\[ \n{\sum }^{ * }\left( s\right) = \left\{ {\alpha \in {\mathbb{N}}^{\mathbb{N}} : \left( {\forall i < k}\right) \left( {\alpha \left( i\right) \leq {n}_{i}}\right) }\right\} \n\]\n\nTake any capacity ... | Yes |
Theorem 4.10.12 (Choquet capacitability theorem [30], [107]) Every analytic subset of a Polish space is universally capacitable. | Proof. Let \( X \) be a Polish space and \( A \subseteq X \) analytic. Let \( I \) be any capacity on \( X \) . Suppose \( I\left( A\right) > t \) . Let \( f : {\mathbb{N}}^{\mathbb{N}} \rightarrow X \) be a continuous map with range \( A \) . By 4.10.11, there is a compact \( K \subseteq {\mathbb{N}}^{\mathbb{N}} \) s... | No |
Proposition 4.10.13 Let \( X \) be a Polish space and \( I \) the separation capacity on \( X \times X \) as defined in 4.10.2. Assume that a rectangle \( {A}_{1} \times {A}_{2} \) be universally capacitable. If \( I\left( {{A}_{1} \times {A}_{2}}\right) = 0 \), then there is a Borel rectangle \( B = {B}_{1} \times {B}... | Proof of 4.10.13. Set \( {C}_{0} = {A}_{1} \times {A}_{2} \) . By 4.10.10, there is a Borel \( {C}_{1} \supseteq {C}_{0} \) such that \( I\left( {C}_{1}\right) = 0 \) . Set \( {C}_{2} = R\left\lbrack {C}_{1}\right\rbrack \) . (Recall that \( \mathrm{R}\left\lbrack \mathrm{A}\right\rbrack \) denotes the smallest rectang... | Yes |
Theorem 4.11.1 (Second separation theorem for analytic sets) (Kuratowski) Let \( X \) be a Polish space and \( A, B \) two analytic subsets. There exist disjoint coanalytic sets \( C \) and \( D \) such that\n\n\[ A \smallsetminus B \subseteq C \\text{and} B \smallsetminus A \subseteq D. \] | Proof. By 4.1.20, there exist Borel maps \( f : X \rightarrow {LO}, g : X \rightarrow {LO} \) such that \( {f}^{-1}\left( {WO}\right) = {A}^{c} \) and \( {g}^{-1}\left( {WO}\right) = {B}^{c} \). \n\nFor \( \alpha ,\beta \) in \( {LO} \), define\n\n\[ \alpha \preccurlyeq \beta \; \Leftrightarrow \;\exists f \in {\\mathb... | Yes |
Corollary 4.11.3 Suppose \( X \) is a Polish space and \( \left( {A}_{n}\right) \) a sequence of analytic subsets of \( X \) . Then there exists a sequence \( \left( {C}_{n}\right) \) of pairwise disjoint coanalytic sets such that | Proof. By the second separation theorem, for each \( n \) there exist pairwise disjoint coanalytic sets \( {C}_{n}^{\prime } \) and \( {D}_{n}^{\prime } \) such that\n\n\[ {A}_{n} \smallsetminus \mathop{\bigcup }\limits_{{m \neq n}}{A}_{m} \subseteq {C}_{n}^{\prime }\text{ and }\mathop{\bigcup }\limits_{{m \neq n}}{A}_... | Yes |
Proposition 4.11.4 Suppose \( X \) is a Polish space and \( \left( {A}_{n}\right) \) a sequence of analytic subsets of \( X \) . Then there exists a sequence \( \left( {C}_{n}\right) \) of coanalytic subsets of \( X \) such that\n\n\[ \n{A}_{n} \smallsetminus \lim \sup {A}_{m} \subseteq {C}_{n} \n\]\n\n(1)\n\nand\n\n\[... | Proof. For each \( n \), set \( {\beta }_{n} = {\beta }_{{A}_{n}} \), where \( {\beta }_{{A}_{n}} \) is as defined in 4.11.2. Let\n\n\[ \n{Q}_{nm} = \left\{ {x \in X : {\beta }_{n}\left( x\right) \leq {\beta }_{m}\left( x\right) }\right\} \n\]\n\n\( {Q}_{nm} \) is analytic by 4.11.2. Take\n\n\[ \n{C}_{n} = {\left\lbrac... | Yes |
Theorem 4.12.3 (Lusin[71]) If \( X, Y \) are Polish and \( B \) a Borel subset of \( X \times Y \) such that for every \( x \in X \) the section \( {B}_{x} \) is countable, then \( {\pi }_{X}\left( B\right) \) is Borel. | Proof. Let \( E \subseteq {\mathbb{N}}^{\mathbb{N}} \) be a closed set and \( f : E \rightarrow X \times Y \) a one-to-one continuous map from \( E \) onto \( B \) . Consider \( g = {\pi }_{X} \circ f \) . For every \( x \in {\pi }_{X}\left( B\right) \) , \( {g}^{-1}\left( x\right) \) is a countable closed subset of \(... | Yes |
Theorem 4.12.4 Suppose \( X, Y \) are Polish spaces and \( f : X \rightarrow Y \) is a countable-to-one Borel map. Then \( f\left( B\right) \) is Borel for every Borel set \( B \) in \( X \) . | Proof. The result follows from 4.12.3 and the identity\n\n\[ f\left( B\right) = {\pi }_{Y}\left( {\operatorname{graph}\left( f\right) \bigcap \left( {B \times Y}\right) }\right) .\n\] | Yes |
Theorem 4.12.5 (Purves [93]) Let \( X \) be a standard Borel space, \( Y \) Polish, and \( f : X \rightarrow Y \) a bimeasurable map. Then\n\n\[ \left\{ {y \in Y : {f}^{-1}\left( y\right) \text{ is uncountable }}\right\} \] \n\nis countable. | Proof of 4.12.5. Assume that \( {f}^{-1}\left( y\right) \) is uncountable for uncountably many \( y \) . We shall show that there is a Borel \( B \subseteq X \) such that \( f\left( B\right) \) is not Borel.\n\nCase 1: \( f \) is continuous.\n\nFix a countable base \( \left( {U}_{n}\right) \) for the topology of \( X \... | Yes |
Lemma 4.12.6 Let \( X \) be a standard Borel space, \( Y \) Polish, and \( A \subseteq X \times \) \( Y \) analytic with \( {\pi }_{X}\left( A\right) \) uncountable. Suppose that for every \( x \in {\pi }_{X}\left( A\right) \) , the section \( {A}_{x} \) is perfect. Then there is a \( C \subseteq {\pi }_{X}\left( A\rig... | Proof of 4.12.6.\n\nFix a compatible complete metric on \( Y \) and a countable base \( \left( {U}_{n}\right) \) for the topology of \( Y \) . For each \( s \in {2}^{ < \mathbb{N}} \), we define a map \( {n}_{s}\left( x\right) : {\pi }_{X}\left( A\right) \rightarrow \mathbb{N} \) satifying the following conditions.\n\n... | No |
Lemma 5.1.2 Suppose \( Y \) is metrizable, \( G : X \rightarrow Y \) strongly \( \mathcal{A} \) - measurable, and \( \mathcal{A} \) closed under countable unions. Then \( G \) is \( \mathcal{A} \) -measurable. | Proof. Let \( U \) be open in \( Y \) . Since \( Y \) is metrizable, \( U \) is an \( {F}_{\sigma } \) set in \( Y \) . Let \( U = \mathop{\bigcup }\limits_{n}{C}_{n},{C}_{n} \) closed. Then\n\n\[ \n{G}^{-1}\left( U\right) = \mathop{\bigcup }\limits_{n}{G}^{-1}\left( {C}_{n}\right) \n\]\n\nSince \( G \) is strongly \( ... | Yes |
Lemma 5.1.4 Suppose \( \left( {X,\mathcal{A}}\right) \) is a measurable space, \( Y \) a Polish space, and \( G : X \rightarrow Y \) a closed-valued measurable multifunction. Then \( \operatorname{gr}\left( G\right) \in \) \( \mathcal{A} \otimes {\mathcal{B}}_{Y} \) | Proof. Let \( \left( {U}_{n}\right) \) be a countable base for \( Y \) . Note that\n\n\[ y \notin G\left( x\right) \Leftrightarrow \exists n\left\lbrack {G\left( x\right) \bigcap {U}_{n} = \varnothing \& y \in {U}_{n}}\right\rbrack . \]\n\nTherefore,\n\n\[ \left( {X \times Y}\right) \smallsetminus \operatorname{gr}\lef... | Yes |
Proposition 5.1.9 Suppose \( X \) is a Polish space and \( \mathbf{\Pi } \) a Borel equivalence relation on \( X \) . Then the following statements are equivalent.\n\n(i) \( \Pi \) has a Borel section.\n\n(ii) II admits a Borel cross section. | Proof. If \( f \) is a Borel section of \( \mathbf{\Pi } \), then the corresponding cross section is clearly Borel. On the other hand, let \( S \) be a Borel cross section of \( \mathbf{\Pi } \) . Let \( f\left( x\right) \) be the unique point of \( S \) equivalent to \( x \) . It is clearly a section of \( \mathbf{\Pi... | Yes |
Proposition 5.1.11 Every closed equivalence relation \( \mathbf{\Pi } \) on a Polish space \( X \) is countably separated. | Proof. Take any countable base \( \left( {U}_{n}\right) \) for the topology of \( X \) . For every \( x, y \) in \( X \) such that \( \left( {x, y}\right) \notin \mathbf{\Pi } \), there exist basic open sets \( {U}_{n} \) and \( {U}_{m} \) containing \( x \) and \( y \) respectively with \( {U}_{n} \times {U}_{m} \subs... | Yes |
Proposition 5.1.12 Every Borel measurable partition of a Polish space into \( {G}_{\delta } \) sets is countably separated. | Proof. Let \( X \) be a Polish space and \( \mathbf{\Pi } \) a Borel measurable partition of \( X \) into \( {G}_{\delta } \) sets. Take \( Y = F\left( X\right) \), the Effros Borel space of \( X \) . Then \( Y \) is standard Borel (3.3.10). For \( x \in X \), let \( \left\lbrack x\right\rbrack \) be the equivalence cl... | Yes |
Lemma 5.1.16 Let \( \Pi \) be a Borel partition of a Polish space \( X \) . The following statements are equivalent.\n\n(i) \( \Pi \) is countably separated.\n\n(ii) The \( \sigma \) -algebra \( {\mathcal{B}}^{ * } \) of \( \mathbf{\Pi } \) -invariant Borel sets is countably generated. | Proof. (i) implies (ii): Let \( \mathbf{\Pi } \) be countably separated. Take a Polish space \( Y \) and \( f : X \rightarrow Y \) a Borel map such that\n\n\[ x \coprod {x}^{\prime } \Leftrightarrow f\left( x\right) = f\left( {x}^{\prime }\right) .\n\]\n\nWe show that \( {\mathcal{B}}^{ * } = {f}^{-1}\left( {\mathcal{B... | Yes |
Theorem 5.2.1 (Kuratowski and Ryll-Nardzewski [63]) Every \( {\mathcal{L}}_{\sigma } \) - measurable, closed-valued multifunction \( F : X \rightarrow Y \) admits an \( {\mathcal{L}}_{\sigma } \) - measurable selection. | Proof of 5.2.1. Inductively we define a sequence \( \left( {s}_{n}\right) \) of \( {\mathcal{L}}_{\sigma } \) -measurable maps from \( X \) to \( Y \) such that for every \( x \in X \) and every \( n \in \mathbb{N} \) ,\n\n(i) \( d\left( {{s}_{n}\left( x\right), F\left( x\right) }\right) < {2}^{-n} \), and\n\n(ii) \( d... | Yes |
Lemma 5.2.2 Suppose \( {A}_{n} \in {\mathcal{L}}_{\sigma } \). Then there exist \( {B}_{n} \subseteq {A}_{n} \) such that the \( {B}_{n} \)’s are pairwise disjoint elements of \( {\mathcal{L}}_{\sigma } \) and \( \mathop{\bigcup }\limits_{n}{A}_{n} = \mathop{\bigcup }\limits_{n}{B}_{n} \). | Proof. Write\n\n\[ \n{A}_{n} = \mathop{\bigcup }\limits_{m}{C}_{nm} \n\] \n\n\( {C}_{nm} \in \mathcal{L} \). Enumerate \( \left\{ {{C}_{nm} : n, m \in \mathbb{N}}\right\} \) in a single sequence, say \( \left( {D}_{i}\right) \). Set\n\n\[ \n{E}_{i} = {D}_{i} \smallsetminus \mathop{\bigcup }\limits_{{j < i}}{D}_{j} \n\]... | Yes |
Lemma 5.2.3 Suppose \( {f}_{n} : X \rightarrow Y \) is a sequence of \( {\mathcal{L}}_{\sigma } \) -measurable functions converging uniformly to \( f : X \rightarrow Y \) . Then \( f \) is \( {\mathcal{L}}_{\sigma } \) -measurable. | Proof. Replacing \( \left( {f}_{n}\right) \) by a subsequence if necessary, we assume that\n\n\[ \forall x\forall n\left( {d\left( {f\left( x\right) ,{f}_{n}\left( x\right) }\right) < 1/\left( {n + 1}\right) }\right) .\n\]\n\nLet \( F \) be a closed set in \( Y \) and\n\n\[ {F}_{n} = \operatorname{cl}\left( {\{ y \in Y... | Yes |
Corollary 5.2.4 Let \( X \) be a Polish space and \( F\left( X\right) \) the space of nonempty closed subsets of \( X \) with Effros Borel structure. Then there is a measurable \( s : F\left( X\right) \rightarrow X \) such that \( s\left( F\right) \in F \) for all \( F \in F\left( X\right) \) . | Proof. Apply 5.2.1 to the multifunction \( G : F\left( X\right) \rightarrow X \), where \( G\left( F\right) = \) \( F \), with \( \mathcal{L} \) the Effros Borel \( \sigma \) -algebra on \( F\left( X\right) \) . | Yes |
Corollary 5.2.5 Let \( \\left( {T,\\mathcal{T}}\\right) \) be a measurable space and \( Y \) a separable metric space. Then every \( \\mathcal{T} \) -measurable, compact-valued multifunction \( F \) : \( T \\rightarrow Y \) admits a \( \\mathcal{T} \) -measurable selection. | Proof. Let \( X \) be the completion of \( Y \) . Then \( F \) as a multifunction from \( T \) to \( X \) is closed-valued and \( \\mathcal{T} \) -measurable. Apply 5.2.1 now. | No |
Corollary 5.2.6 Suppose \( Y \) is a compact metric space, \( X \) a metric space, and \( f : Y \rightarrow X \) a continuous onto map. Then there is a Borel map \( s : X \rightarrow Y \) of class 2 such that \( f \circ s \) is the identity map on \( X \) . | Proof. Let \( F\left( x\right) = {f}^{-1}\left( x\right), x \in X \), and \( \mathcal{L} = {\mathbf{\Delta }}_{2}^{0} \) . For any closed set \( C \) in \( Y \) ,\n\n\[ \n{F}^{-1}\left( C\right) = {\pi }_{X}\left( {\operatorname{graph}\left( f\right) \bigcap \left( {X \times C}\right) }\right) .\n\]\n\nTherefore, by \(... | Yes |
Proposition 5.2.7 (A. Maitra and B. V. Rao[77]) Let \( T \) be a nonempty set, \( \mathcal{L} \) an algebra on \( T \), and \( X \) a Polish space. Suppose \( F : T \rightarrow X \) is a closed-valued \( {\mathcal{L}}_{\sigma } \) -measurable multifunction. Then there is a sequence \( \left( {f}_{n}\right) \) of \( {\m... | Proof. Fix a countable base \( \left\{ {{U}_{n} : n \in \mathbb{N}}\right\} \) for the topology of \( X \) and fix also an \( {\mathcal{L}}_{\sigma } \) -measurable selection \( f \) for \( F \) . For each \( n,{T}_{n} = {F}^{-1}\left( {U}_{n}\right) \in {\mathcal{L}}_{\sigma } \) . Write \( {T}_{n} = \mathop{\bigcup }... | Yes |
Theorem 5.2.8 (Srivastava[115]) Let \( T,\mathcal{L}, X \), and \( F \) be as in 5.2.7. Then there is a map \( f : T \times {\mathbb{N}}^{\mathbb{N}} \rightarrow X \) such that\n\n(i) for every \( \alpha \in {\mathbb{N}}^{\mathbb{N}}, t \rightarrow f\left( {t,\alpha }\right) \) is \( {\mathcal{L}}_{\sigma } \) -measura... | Proof of 5.2.8 Fix a complete compatible metric \( d \) on \( X \) . Applying 5.2.9 and 5.2.7 repeatedly, for each \( s \in {\mathbb{N}}^{ < \mathbb{N}} \), we get an \( {\mathcal{L}}_{\sigma } \) -measurable selection \( {f}_{s} : T \rightarrow X \) for \( F \) satisfying the following condition: For every \( s \in {\... | Yes |
Theorem 5.2.10 (S. Bhattacharya and S. M. Srivastava [12]) Let \( F \) : \( X \rightarrow Y \) be closed-valued and strongly \( {\mathcal{L}}_{\sigma } \) -measurable. Suppose \( Z \) is a separable metric space and \( g : Y \rightarrow Z \) a Borel map of class 2 . Then there is an \( {\mathcal{L}}_{\sigma } \) -measu... | Proof. Let \( \left( {U}_{n}\right) \) be a countable base for the topology of \( Z \) . Write \( {g}^{-1}\left( {U}_{n}\right) = \mathop{\bigcup }\limits_{m}{H}_{nm} \), the \( {H}_{nm} \) ’s closed. Also, take a countable base \( \left( {W}_{n}\right) \) for \( Y \) and write \( {W}_{n} = \mathop{\bigcup }\limits_{m}... | Yes |
Theorem 5.2.11 Let \( X, Y \) be compact metric spaces, \( f : X \rightarrow Y \) a continuous onto map. Suppose \( A \subseteq Y \) and \( 1 \leq \alpha < {\omega }_{1} \). Then\n\n\[ \n{f}^{-1}\left( A\right) \in {\mathbf{\Pi }}_{\alpha }^{0}\left( X\right) \Leftrightarrow A \in {\mathbf{\Pi }}_{\alpha }^{0}\left( Y\... | Proof of 5.2.11 We need to prove the \ | No |
Lemma 5.2.12 Let \( X, Y \), and \( f \) be as in 5.2.11. Suppose \( 1 \leq \alpha < {\omega }_{1}, Z \) is a separable metric space, and \( g : X \rightarrow Z \) is a Borel map of class \( \alpha \) . Then there is a class 2 map \( s : Y \rightarrow X \) such that \( g \circ s \) is of class \( \alpha \) and \( f\lef... | Proof. Let \( F\left( y\right) = {f}^{-1}\left( y\right), y \in Y \) . Then \( F : Y \rightarrow X \) is an upper-semicontinuous closed-valued multifunction. By 5.2.1 there is a selection \( s \) of \( F \) that is Borel of class 2 . This \( s \) works if either \( \alpha = 1 \) (i.e., if \( g \) is continuous) or if \... | Yes |
Theorem 5.3.1 (Schäl) Suppose \( \\left( {T,\\mathcal{T}}\\right) \) is a measurable space and let \( Y \) be a separable metric space. Suppose \( G : T \\rightarrow Y \) is a \( \\mathcal{T} \) -measurable compact-valued multifunction. Let \( v \) be a real-valued function on \( \\operatorname{gr}\\left( G\\right) \) ... | Proof of 5.3.1. (Burgess and Maitra[24]) Without any loss of generality we assume that \( Y \) is Polish. Fix a complete metric \( d \) on \( Y \) compatible with its topology. By 5.2.7, we get \( \\mathcal{T} \) -measurable selections \( {g}_{n} : T \\rightarrow Y \) of \( G \) such that \n\n\[ \nG\\left( t\\right) = ... | Yes |
It is not unreasonable to conjecture that 5.3.1 remains true even for \( v \) that are \( \mathcal{T}\bigotimes {\mathcal{B}}_{Y} \mid {gr}\left( G\right) \) -measurable such that \( v\left( {t,\text{.}}\right) {isupper} \) semicontinuous for every \( t \) . However, this is not true. | Recall that in the last chapter, using Solovay's coding of Borel sets, we showed that there is a coanalytic set \( T \) and a function \( g : T \rightarrow {2}^{\mathbb{N}} \) whose graph is relatively Borel in \( T \times {2}^{\mathbb{N}} \) but that is not Borel measurable. Take \( \mathcal{T} = {\mathcal{B}}_{T}, G\... | Yes |
Theorem 5.4.1 (Effros [40]) Every lower-semicontinuous or upper-semicontinuous partition \( \mathbf{\Pi } \) of a Polish space \( X \) into closed sets admits a Borel measurable section \( f : X \rightarrow X \) of class 2. In particular, they admit a \( {G}_{\delta } \) cross section. | Proof. In 5.2.1, take \( Y = X,\mathcal{L} \) the family of invariant sets that are simultaneously \( {F}_{\sigma } \) and \( {G}_{\delta } \), and \( F\left( x\right) = \left\lbrack x\right\rbrack \), the equivalence class containing \( x \) . So, there is an \( {\mathcal{L}}_{\sigma } \) -measurable selection \( f : ... | Yes |
Theorem 5.4.2 (Effros - Mackey cross section theorem) Suppose \( H \) is a closed subgroup of a Polish group \( G \) and \( \mathbf{\Pi } \) the partition of \( G \) consisting of all the right cosets of \( H \) . Then \( \mathbf{\Pi } \) admits a Borel measurable section of class 2. In particular, it admits a \( {G}_{... | Proof. Note that for any open set \( U \) in \( G \) ,\n\n\[ \n{U}^{ * } = \bigcup \{ g \cdot U : g \in H\} . \n\]\n\nSo, \( {U}^{ * } \) is open. Thus \( \mathbf{\Pi } \) is lower semicontinuous. The result follows from Effros's cross section theorem (5.4.1). | No |
Theorem 5.4.3 Every Borel measurable partition \( \mathbf{\Pi } \) of a Polish space \( X \) into closed sets admits a Borel measurable section \( f : X \rightarrow X \) . In particular, it admits a Borel cross section. | Proof. Let \( \mathcal{A} \) be the \( \sigma \) -algebra of all invariant Borel subsets of \( X \) and \( F : X \rightarrow X \) the multifunction that assigns to each \( x \in X \) the member of \( \mathbf{\Pi } \) containing \( x \) . By our assumptions, \( F \) is \( \mathcal{A} \) -measurable. By 5.2.1, we get a m... | Yes |
Theorem 5.4.4 The classification space \( \operatorname{irr}\left( n\right) / \sim \) is standard Borel. | Proof. Fix any irreducible \( A \) . Then the \( \sim \) -equivalence class \( \left\lbrack A\right\rbrack \) containing \( A \) equals\n\n\[ \n{\pi }_{1}\left\{ {\left( {B, U}\right) \in \operatorname{irr}\left( n\right) \times U\left( n\right) : A = {UB}{U}^{ * }}\right\} , \n\]\n\nwhere \( {\pi }_{1} : \operatorname... | Yes |
Theorem 5.4.5 (Miller[84]) Let \( \\left( {G, \\cdot }\\right) \) be a Polish group, \( X \) a Polish space, and \( a\\left( {g, x}\\right) = g \\cdot x \) an action of \( G \) on \( X \) . Suppose for a given \( x \\in X \) that \( g \\rightarrow g \\cdot x \) is Borel. Then the orbit\n\n\\[ \n\\{ g \\cdot x : g \\in ... | Proof. Let \( H = {G}_{x} \) be the stabilizer of \( x \) . By 4.8.4, \( H \) is closed in \( G \) . Let \( S \) be a Borel cross section of the partition \( \\mathbf{\\Pi } \) consisting of the left cosets of \( H \) . The map \( g \\rightarrow g \\cdot x \) restricted to \( S \) is one-to-one, Borel, and onto the orb... | Yes |
Proposition 5.5.1 Let \( X, Y \) be Polish spaces, \( B \subseteq X \times Y \) Borel, and \( C \) an analytic uniformization of \( B \) . Then \( C \) is Borel. | Proof. We show that \( C \) is also coanalytic. The result will then follow from Souslin’s theorem. That \( C \) is coanalytic follows from the following relation:\n\n\[ \left( {x, y}\right) \in C \Leftrightarrow \left( {x, y}\right) \in B\& \forall z\left( {\left( {x, z}\right) \in C \Rightarrow y = z}\right) . \] | No |
Theorem 5.5.2 (Von Neumann[124]) Let \( X \) and \( Y \) be Polish spaces, \( A \subseteq \) \( X \times Y \) analytic, and \( \mathcal{A} = \sigma \left( {{\mathbf{\sum }}_{1}^{1}\left( X\right) }\right) \), the \( \sigma \) -algebra generated by the analytic subsets of \( X \) . Then there is an \( \mathcal{A} \) -me... | Proof. Let \( f : {\mathbb{N}}^{\mathbb{N}} \rightarrow A \) be a continuous surjection. Consider\n\n\[ B = \left\{ {\left( {x,\alpha }\right) \in X \times {\mathbb{N}}^{\mathbb{N}} : {\pi }_{X}\left( {f\left( \alpha \right) }\right) = x}\right\} . \]\n\nThen \( B \) is a closed set with \( {\pi }_{X}\left( B\right) = ... | Yes |
Theorem 5.5.3 Every analytic subset \( A \) of the product of Polish spaces \( X, Y \) admits a section \( u \) that is universally measurable as well as Baire measurable. | Proof. The result follows from 5.5.2, 4.3.1, and 4.3.2. | No |
Proposition 5.5.4 In 5.5.3, further assume that \( A \) is Borel. Then the graph of the section \( u \) is coanalytic. | Proof. Note that\n\n\[ \begin{matrix} u\left( x\right) = y & \Leftrightarrow & \left( {x, y}\right) \in A\;\& \;\left( {\forall \alpha \in {\mathbb{N}}^{\mathbb{N}}}\right) \left( {\forall \beta \in {\mathbb{N}}^{\mathbb{N}}}\right) (\lbrack \left( {x,\alpha }\right) \in B \end{matrix}\n\n\[ \left. {\& \left( {x,\beta ... | No |
Theorem 5.5.7 Let \( \left( {X,\mathcal{E}}\right) \) be a measurable space with \( \mathcal{E} \) closed under the Souslin operation, \( Y \) a Polish space, and \( A \in \mathcal{E}\bigotimes {\mathcal{B}}_{Y} \) . Then \( {\pi }_{X}\left( A\right) \in \mathcal{E} \) , and there is an \( \mathcal{E} \) -measurable se... | Proof. By 3.1.7, there exists a countable sub \( \sigma \) -algebra \( \mathcal{D} \) of \( \mathcal{E} \) such that \( A \in \mathcal{D}\bigotimes {\mathcal{B}}_{Y} \) . Let \( \left( {B}_{n}\right) \) be a countable generator of \( \mathcal{D} \) and \( \chi : X \rightarrow \mathcal{C} \) the map defined by\n\n\[ \ch... | Yes |
Theorem 5.5.7 Let \( \\left( {X,\\mathcal{E}}\\right) \) be a measurable space with \( \\mathcal{E} \) closed under the Souslin operation, \( Y \) a Polish space, and \( A \\in \\mathcal{E}\\bigotimes {\\mathcal{B}}_{Y} \) . Then \( {\\pi }_{X}\\left( A\\right) \\in \\mathcal{E} \) , and there is an \( \\mathcal{E} \) ... | Proof. By 3.1.7, there exists a countable sub \( \\sigma \) -algebra \( \\mathcal{D} \) of \( \\mathcal{E} \) such that \( A \\in \\mathcal{D}\\bigotimes {\\mathcal{B}}_{Y} \) . Let \( \\left( {B}_{n}\\right) \) be a countable generator of \( \\mathcal{D} \) and \( \\chi : X \\rightarrow \\mathcal{C} \) the map defined... | Yes |
Corollary 5.5.8 Let \( \left( {X,\mathcal{A}, P}\right) \) be a complete probability space, \( Y \) a Polish space, and \( B \in \mathcal{A}\bigotimes {\mathcal{B}}_{Y} \) . Then \( {\pi }_{X}\left( B\right) \in \mathcal{A} \), and \( B \) admits an \( \mathcal{A} \) -measurable section. | Proof. Since \( \mathcal{A} \) is closed under the Souslin operation, the result follows from 5.5.7. | No |
Theorem 5.7.1 (Novikov [90]) Let \( X, Y \) be Polish spaces and \( \mathcal{A} \) a countably generated sub \( \sigma \) -algebra of \( {\mathcal{B}}_{X} \) . Suppose \( B \in \mathcal{A}\bigotimes {\mathcal{B}}_{Y} \) is such that the sections \( {B}_{x} \) are compact. Then \( {\pi }_{X}\left( B\right) \in \mathcal{... | Proof. Since the projection of a Borel set with compact sections is Borel (4.7.11), \( {\pi }_{X}\left( B\right) \) is Borel. Since \( {\pi }_{X}\left( B\right) \) is a union of atoms of \( \mathcal{A} \), by the Blackwell - Mackey theorem (4.5.7), it is in \( \mathcal{A} \) .\n\nLet \( U \) be an open set in \( Y \) .... | Yes |
Theorem 5.7.2 (Lusin) Let \( X, Y \) be Polish spaces and \( B \subseteq X \times Y \) Borel with sections \( {B}_{x} \) countable. Then \( B \) admits a Borel uniformization. | Proof. By 3.3.17, there is a closed set \( E \) in \( {\mathbb{N}}^{\mathbb{N}} \) and a one-to-one continuous map \( f : E \rightarrow X \times Y \) with range \( B \) . Set\n\n\[ H = \left\{ {\left( {x,\alpha }\right) \in X \times E : {\pi }_{X}\left( {f\left( \alpha \right) }\right) = x}\right\} . \]\n\nThen \( H \)... | Yes |
Proposition 5.7.3 Let \( X \) be a Polish space and \( \Pi \) a countably separated partition of \( X \) with all equivalence classes countable. Then \( \mathbf{\Pi } \) admits a Borel cross section. | Proof. Let \( Y \) be a Polish space and \( f : X \rightarrow Y \) a Borel map such that\n\n\[ \n{x\Pi }{x}^{\prime } \Leftrightarrow f\left( x\right) = f\left( {x}^{\prime }\right) .\n\]\n\nDefine\n\n\[ \nB = \{ \left( {y, x}\right) \in Y \times X : f\left( x\right) = y\} .\n\]\n\nThen \( B \) is a Borel set with sect... | Yes |
Theorem 5.8.4 (Kechris [52]) Let \( X, Y \) be Polish spaces. Assume that \( x \rightarrow {\mathcal{I}}_{x} \) is a Borel on Borel map assigning to each \( x \in X \) a \( \sigma \) -ideal \( {\mathcal{I}}_{x} \) of subsets of \( Y \) . Suppose \( B \subseteq X \times Y \) is a Borel set such that for every \( x \in {... | Proof. Since \( x \rightarrow {\mathcal{I}}_{x} \) is Borel on Borel,\n\n\[{\pi }_{X}\left( B\right) = {\left\{ x : {B}_{x} \in {\mathcal{I}}_{x}\right\} }^{c}\]\n\nis Borel.\n\nIt remains to prove that \( B \) admits a Borel section. Fix a closed subset \( F \) of \( {\mathbb{N}}^{\mathbb{N}} \) and a continuous bijec... | Yes |
Example 5.8.3 Let \( X, Y \) be Polish spaces and \( G : X \rightarrow Y \) a closed-valued Borel measurable multifunction. Define \( \mathcal{I} : X \rightarrow \mathcal{P}\left( {\mathcal{P}\left( Y\right) }\right) \) by \[ \mathcal{I}\left( x\right) = \{ I \subseteq Y : I\text{ is meager in }G\left( x\right) \} . | By imitating the proof of 3.5.18 we can show the following: For every open set \( U \) in \( Y \) and every Borel set \( B \) in \( X \times Y \), the sets \[ {B}^{*U} = \left\{ {x \in X : G\left( x\right) \bigcap U \neq \varnothing }\right. \] \[ \text{&}{B}_{x}\bigcap G\left( x\right) \bigcap U\text{is comeager in}G\... | No |
Theorem 5.8.5 (Kechris [52] and Sarbadhikari [100]) If B is a Borel subset of the product of two Polish spaces \( X \) and \( Y \) such that \( {B}_{x} \) is nonmeager in \( Y \) for every \( x \in {\pi }_{X}\left( B\right) \), then \( B \) admits a Borel uniformization. | Proof. Apply 5.8.4 with \( {\mathcal{I}}_{x} \) as in example 5.8.2. | No |
Example 5.8.6 As a special case of 5.8.5 we see that every Borel set \( B \subseteq X \times Y \) with \( {B}_{x} \) a dense \( {G}_{\delta } \) set admits a Borel uniformization. However, there is an \( {F}_{\sigma } \) subset \( E \) of \( \left\lbrack {0,1}\right\rbrack \times {\mathbb{N}}^{\mathbb{N}} \) with secti... | Let \( C \subseteq \left\lbrack {0,1}\right\rbrack \times {\mathbb{N}}^{\mathbb{N}} \) be a closed set with projection to the first coordinate space \( \left\lbrack {0,1}\right\rbrack \), that does not admit a Borel uniformization. Such a set exists by 5.1.7. For each \( s \in {\mathbb{N}}^{ < \mathbb{N}} \), fix a hom... | No |
Theorem 5.8.7 (Blackwell and Ryll-Nardzewski [17]) Let \( X, Y \) be Polish spaces, \( P \) a transition probability on \( X \times Y \), and \( B \) a Borel subset of \( X \times Y \) such that \( P\left( {x,{B}_{x}}\right) > 0 \) for all \( x \in {\pi }_{X}\left( B\right) \) . Then \( {\pi }_{X}\left( B\right) \) is ... | Proof. Apply 5.8.4 with \( {\mathcal{I}}_{x} \) as in Example 5.8.1. | No |
Theorem 5.8.8 (Blackwell and Ryll-Nardzewski) Let \( X, Y \) be Polish spaces, \( \mathcal{A} \) a countably generated sub \( \sigma \) algebra of \( {\mathcal{B}}_{X} \), and \( P \) a transition probability on \( X \times Y \) such that for every \( B \in {\mathcal{B}}_{Y}, x \rightarrow P\left( {x, B}\right) \) is \... | Proof of 5.8.8. By a slight modification of the argument contained in the proof of 3.4.24 we see that for every \( E \in \mathcal{A} \otimes {\mathcal{B}}_{Y}, x \rightarrow P\left( {x,{E}_{x}}\right) \) is \( \mathcal{A} \) -measurable. As \( {\pi }_{X}\left( B\right) = \left\{ {x \in X : P\left( {x,{B}_{x}}\right) > ... | Yes |
Lemma 5.8.9 Let \( X, Y,\mathcal{A} \), and \( P \) be as above. For every \( E \in \mathcal{A}\bigotimes {\mathcal{B}}_{Y} \) and every \( \epsilon > 0 \), there is an \( F \in \mathcal{A}\bigotimes {\mathcal{B}}_{Y} \) contained in \( E \) such that \( {F}_{x} \) is compact and \( P\left( {x,{F}_{x}}\right) \geq \eps... | Proof. Let \( \mathcal{M} \) be the class of all sets in \( \mathcal{A}\bigotimes {\mathcal{B}}_{Y} \) such that the conclusion of the lemma holds for every \( P \) and every \( \epsilon > 0 \) . By 3.4.20, \( \mathcal{M} \) contains all rectangles \( A \times B \), where \( A \in \mathcal{A} \) and \( B \) Borel in \(... | Yes |
Proposition 5.8.10 Let \( X, f \), and \( \mathcal{A} \) be as above. An everywhere proper conditional distribution given \( f \) exists if and only if there is an \( \mathcal{A} \) - measurable \( g : X \rightarrow X \) such that \( f\left( {g\left( x\right) }\right) = f\left( x\right) \) for all \( x \) . | Proof. Suppose an \( \mathcal{A} \) -measurable \( g : X \rightarrow X \) such that \( f \circ g \) is the identity exists. Define\n\n\[ Q\left( {x, B}\right) = \left\{ \begin{array}{ll} 1 & \text{ if }g\left( x\right) \in B \\ 0 & \text{ otherwise. } \end{array}\right. \]\n\nIt is easy to verify that \( Q \) has the d... | Yes |
Proposition 5.8.13 (Feldman and Moore [41]) Every Borel equivalence relation on a Polish space \( X \) with equivalence classes countable is induced by a countable group of Borel automorphisms. | Proof. Let \( \Pi \) be a Borel equivalence relation on \( X \) with equivalence classes countable. By 5.8.11, write\n\n\[ \Pi = \mathop{\bigcup }\limits_{n}{G}_{n} \]\n\nwhere \( {\pi }_{1} \mid {G}_{n} \) is one-to-one, \( {\pi }_{1}\left( {x, y}\right) = x \) ; i.e., the \( {G}_{n} \) ’s are graphs of Borel function... | Yes |
Theorem 5.9.1 (Miller [85]) Every partition \( \Pi \) of a Polish space \( X \) into \( {G}_{\delta } \) sets such that the saturation of every basic open set is simultaneously \( {F}_{\sigma } \) and \( {G}_{\delta } \) admits a section \( s : X \rightarrow X \) that is Borel measurable of class 2. In particular, such... | Proof. Let \( \left( {U}_{n}\right) \) be a countable base for the topology of \( X \) . Let \( \left( {V}_{n}\right) \) be an enumeration of \( \left\{ {{U}_{n}^{ * } : n \in \mathbb{N}}\right\} \bigcup \left\{ {{\left( {U}_{n}^{ * }\right) }^{c} : n \in \mathbb{N}}\right\} \) . Let \( {\mathcal{T}}^{\prime } \) be th... | Yes |
Theorem 5.9.2 (Srivastava [114]) Every Borel measurable partition \( \mathbf{\Pi } \) of a Polish space \( X \) into \( {G}_{\delta } \) sets admits a Borel cross section. | Proof. (Kechris) For \( x \in X \) let \( \left\lbrack x\right\rbrack \) denote the member of \( \mathbf{\Pi } \) containing \( x \) . Consider the multifunction \( p : X \rightarrow X \) defined by\n\n\[ p\left( x\right) = \operatorname{cl}\left( \left\lbrack x\right\rbrack \right) \]\n\nThen \( p : X \rightarrow X \)... | Yes |
Theorem 5.9.5 \( \operatorname{irr}\left( A\right) / \sim \) is standard Borel if and only if \( A \) is GCR. | Its proof makes crucial uses of 5.4.3 and 4.5.4. We refer the interested reader to [4] and [43] for a proof. | No |
Theorem 5.10.1 (The reflection theorem) Let \( X \) be a Polish space and \( \Phi \subseteq \mathcal{P}\left( X\right) {\mathbf{\Pi }}_{1}^{1} \) on \( {\mathbf{\Pi }}_{1}^{1} \) . For every \( {\mathbf{\Pi }}_{1}^{1} \) set \( A \in \Phi \) there is a Borel \( B \subseteq A \) in \( \Phi \) . | Proof. Suppose there is a \( {\mathbf{\Pi }}_{1}^{1} \) set \( A \subseteq X \) in \( \Phi \) that does not contain a Borel set belonging to \( \Phi \) . We shall get a contradiction. Let \( \varphi \) be a \( {\mathbf{\Pi }}_{1}^{1} \) -norm on \( A \) and\n\n\[ C = \left\{ {\left( {x, y}\right) : y{ < }_{\varphi }^{ ... | Yes |
Theorem 5.10.2 Let \( X, Y \) be Polish spaces and \( A \subseteq X \times Y \) analytic with sections \( {A}_{x} \) countable. Then every coanalytic set \( B \) containing \( A \) contains a Borel set \( E \supseteq A \) with all sections countable. | Proof. Let \( C = {B}^{c} \) . Define \( \Phi \subseteq \mathcal{P}\left( {X \times Y}\right) \) by\n\n\[ D \in \Phi \Leftrightarrow {D}^{c} \subseteq B\& \forall x\left( {\left( {D}^{c}\right) }_{x}\right. \text{is countable})\text{.}\]\n\nUsing 4.3.7 we can easily check that \( \Phi \) is \( {\mathbf{\Pi }}_{1}^{1} \... | Yes |
Theorem 5.10.3 (Lusin) Every analytic set with countable sections, in the product of two Polish spaces, can be covered by countably many Borel graphs. | Proof. The result immediately follows from 5.10.2 and 5.8.11. | No |
Proposition 5.10.4 (Burgess) Let \( X \) be Polish, \( E \) an analytic equivalence relation on \( X \), and \( C \subseteq X \times X \) a coanalytic set containing \( E \) . Then there is a Borel equivalence relation \( B \) such that \( E \subseteq B \subseteq C \) . | Proof of 5.10.4. Applying 5.10.5 repeatedly, by induction on \( n \) we can define a sequence of Borel sets \( \left( {B}_{n}\right) \) such that\n\n\[ E \subseteq {B}_{n} \subseteq \mathcal{E}\left( {B}_{n}\right) \subseteq {B}_{n + 1} \subseteq C \]\n\nfor all \( n \) . Take \( B = \mathop{\bigcup }\limits_{n}{B}_{n}... | No |
Lemma 5.10.5 Let \( X \) be a Polish space, \( P \) analytic, \( C \) coanalytic, and \( \mathcal{E}\left( P\right) \subseteq C \) . Then there is a Borel set \( B \) containing \( P \) such that\n\n\[ \mathcal{E}\left( B\right) \subseteq C\text{.} \] | Proof. Define \( \Phi \subseteq \mathcal{P}\left( {X \times X}\right) \) by\n\n\[ D \in \Phi \Leftrightarrow \mathcal{E}\left( {D}^{c}\right) \subseteq C. \]\n\n\( \Phi \) is \( {\mathbf{\Pi }}_{1}^{1} \) on \( {\mathbf{\Pi }}_{1}^{1} \) . Further, \( {P}^{c} \in \Phi \) . By the reflection theorem (5.10.1), there is a... | Yes |
Corollary 5.10.6 For every analytic equivalence relation \( E \) on a Polish space \( X \) there exist Borel equivalence relations \( {B}_{\alpha },\alpha < {\omega }_{1} \), such that \( E = \) \( \mathop{\bigcap }\limits_{{\alpha < {\omega }_{1}}}{B}_{\alpha }. \) | Proof. By 4.3.17, write \( E = \mathop{\bigcap }\limits_{{\alpha < {\omega }_{1}}}{C}_{\alpha },{C}_{\alpha } \) coanalytic. By 5.10.4, for each \( \alpha \) there exists a Borel equivalence relation \( {B}_{\alpha } \) such that \( E \subseteq {B}_{\alpha } \subseteq {C}_{\alpha } \) . | Yes |
Theorem 5.11.4 Every countably generated sub \( \sigma \) -algebra of the Borel \( \sigma \) - algebra of a Polish space has a minimal complement. | Proof of 5.11.4. Let \( X \) be Polish and \( \mathcal{C} \) a countably generated sub \( \sigma \) -algebra of \( {\mathcal{B}}_{X} \) . \n\nCase 1. There is a cocountable atom \( A \) of \( \mathcal{C} \) .\n\nLet \( f : X \smallsetminus A \rightarrow A \) be a one-to-one map. Take\n\n\[ \mathcal{D} = \sigma \left( {... | Yes |
Lemma 5.11.5 Let \( X \) be Polish and \( \mathcal{C} \) a countably generated sub \( \sigma \) -algebra of \( {\mathcal{B}}_{X} \) . Suppose \( \mathcal{D} \) is a countably generated sub \( \sigma \) -algebra of \( {\mathcal{B}}_{X} \) such that every atom \( A \) of \( \mathcal{D} \) is a partial cross section of th... | Proof. Under the hypothesis, \( \mathcal{C} \vee \mathcal{D} \) is a countably generated sub \( \sigma \) - algebra of \( {\mathcal{B}}_{X} \) with atoms singletons. Hence, by 4.5.7, \( \mathcal{C} \vee \mathcal{D} = {\mathcal{B}}_{X} \) . Let \( {\mathcal{D}}^{ * } \) be a proper countably generated sub \( \sigma \) -... | Yes |
Theorem 5.12.1 (Arsenin, Kunugui [60]) Let \( B \subseteq X \times Y \) be a Borel set, \( X, Y \) Polish, such that \( {B}_{x} \) is \( \sigma \) -compact for every \( x \) . Then \( {\pi }_{X}\left( B\right) \) is Borel, and \( B \) admits a Borel uniformization. | Proof of 5.12.1. Write \( B = \mathop{\bigcup }\limits_{n}{B}_{n} \), the \( {B}_{n} \) ’s Borel with compact sections. That this can be done follows from 5.12.3. Then\n\n\[{\pi }_{X}\left( B\right) = \mathop{\bigcup }\limits_{n}{\pi }_{X}\left( {B}_{n}\right)\]\n\nSince the projection of a Borel set with compact secti... | Yes |
Theorem 5.12.3 Let \( X, Y \) be Polish spaces and \( A \subseteq X \times Y \) a Borel set with sections \( {A}_{x} \) \(\sigma\) -compact. Then \( A = \mathop{\bigcup }\limits_{n}{B}_{n} \), where each \( {B}_{n} \) is Borel with \( {\left( {B}_{n}\right) }_{x} \) compact for all \( x \) and all \( n \) . | Proof. The result trivially follows from 5.12.2 by taking \( B = {A}^{c} \) . | No |
Proposition 5.12.4 Let \( B \subseteq X \times Y \) be a Borel set with sections \( {B}_{x} \) that are \( {G}_{\delta } \) sets in \( Y \) . Then there exist Borel sets \( {B}_{n} \) with open sections such that \( B = \mathop{\bigcap }\limits_{n}{B}_{n} \) . | Proof. Let \( Z \) be a compact metric space containing (a homeomorph of) \( Y \) . Then \( B \) is Borel in \( X \times Z \) with sections \( {G}_{\delta } \) sets (2.2.7). By 5.12.3, there exist Borel sets \( {C}_{n} \) in \( X \times Z \) with sections compact such that \( \left( {X \times Z}\right) \smallsetminus B... | Yes |
Proposition 5.12.6 Let \( X \) be a Polish space and \( \mathcal{B} \subseteq F\left( X\right) \) hereditary. Then \( {\Omega }_{{D}_{\mathcal{B}}} = {\mathcal{B}}_{\sigma } \cap F\left( X\right) \) . | Proof. Fix a closed set \( A \subseteq X \) and a countable base \( \left( {U}_{n}\right) \) for \( X \) .\n\nLet \( {D}^{\infty }\left( A\right) = \varnothing \) . Then\n\n\[ A = \mathop{\bigcup }\limits_{{\alpha < {\left| A\right| }_{D}}}\left( {{D}^{\alpha }\left( A\right) \smallsetminus {D}^{\alpha + 1}\left( A\rig... | Yes |
Proposition 5.12.7 Let \( X \) be Polish and \( D \) a derivative on \( X \) such that\n\n\[ \n\\{ \\left( {A, B}\\right) \\in F\\left( X\\right) \\times F\\left( X\\right) : A \\subseteq D\\left( B\\right) \\} \n\]\n\nis analytic. Then\n\n(i) \( {\\Omega }_{D} \) is coanalytic, and\n\n(ii) for every analytic \( \\math... | Proof. Assertion (i) follows from the following equivalence:\n\n\[ \nA \\notin {\\Omega }_{D} \\Leftrightarrow \\exists B\\left( {B \\neq \\varnothing \\& B \\subseteq A\\& B \\subseteq D\\left( B\\right) }\\right) .\n\]\n\n(The sets \( A \) and \( B \) are closed in \( X \).)\n\nSuppose (ii) is false for some analytic... | Yes |
Theorem 5.13.1 (Lopez-Escobar) A subset \( A \) of \( {X}_{L} \) is invariant (with respect to the logic action) and Borel, if and only if there is a sentence \( \sigma \) of \( {L}_{{\omega }_{1}\omega } \) such that \( A = {A}_{\sigma } \) . | Proof. The sufficient part of this result is proved by induction on formulae of \( {L}_{{\omega }_{1}\omega } \) as follows:\n\nFor every formula \( \phi \left\lbrack {{v}_{0},{v}_{1},\ldots ,{v}_{k - 1}}\right\rbrack \), the set\n\n\[ \n{A}_{\phi, k} = \left\{ {\left( {x,{n}_{0},{n}_{1},\ldots ,{n}_{k - 1}}\right) : {... | No |
Theorem 5.13.8 Topological Vaught conjecture holds if \( G \) is a locally compact Polish group. | Assuming 5.13.9, we prove 5.13.8 as follows: Let \( G \) be a locally compact Polish group acting continuously on a Polish space \( X \) . Write \( G = \mathop{\bigcup }\limits_{n}{K}_{n} \) , \( {K}_{n} \) compact. Then, for \( x, y \in X \) ,\n\n\[ \exists g \in G\left( {y = g \cdot x}\right) \Leftrightarrow \exists ... | Yes |
Theorem 5.13.9 Let \( E \) be an analytic equivalence relation on a Polish space \( X \) with all equivalence classes \( {F}_{\sigma } \) . Then the number of equivalence classes is \( \leq {\aleph }_{0} \) or perfectly many. | Proof of 5.13.9. Let \( X \) be a Polish space and \( E \) an analytic equivalence relation on \( X \) with all its equivalence classes \( {F}_{\sigma } \) sets. Further assume that there are uncountably many \( E \) -equivalence classes. Fix a countable base \( \left( {V}_{n}\right) \) for the topology of \( X \) . Le... | No |
Proposition 5.13.10 Suppose \( X \) is a Polish space and \( E \) an equivalence relation on \( X \) which is meager in \( {X}^{2} \) . Then \( E \) has perfectly many equivalence classes. | Proof. Let \( E \subseteq \mathop{\bigcup }\limits_{n}{F}_{n},{F}_{n} \) closed and nowhere dense in \( {X}^{2} \) . Without any loss of generality, we further assume that the diagonal \( \left\{ {\left( {x, y}\right) \in {X}^{2} : x = y}\right\} \) is contained in each of \( {F}_{n} \) .\n\nFor each \( s \in {2}^{ < \... | Yes |
Theorem 5.13.12 (Sami) Topological Vaught conjecture holds if \( G \) is abelian. | Proof. Assume that the number of orbits is uncountable. We shall show that there is a perfect set of inequivalent elements.\n\nLet \( E \) be the equivalence relation on \( X \) defined by\n\n\[ \n{xEy} \Leftrightarrow {G}_{x} = {G}_{y} \n\]\n\nwhere \( {G}_{x} \) is the stabilizer of \( x \) . Let \( y = g \cdot x \) ... | Yes |
Lemma 5.13.14 Suppose \( \left\{ {{A}_{\alpha } : \alpha < {\omega }_{1}}\right\} \) is a family of Borel subsets of a Polish space \( X \) and \( E \) the equivalence relation on \( X \) defined by\n\n\[ \n{xEy} \Leftrightarrow \forall \alpha \left( {x \in {A}_{\alpha } \Leftrightarrow y \in {A}_{\alpha }}\right), x, ... | Proof of 5.13.14. Although the proof of the lemma is messy looking, ideawise it is quite simple. Assume that the number of \( E \) -equivalence classes is \( > {\aleph }_{1} \) . We shall then show that there are perfectly many \( E \) -equivalence classes. The following fact will be used repeatedly in the proof of the... | Yes |
Example 5.13.15 Let \( L \) be a first order language whose non-logical symbols consists of exactly one binary relation symbol. So, \( {X}_{L} = {2}^{\omega \times \omega } \) . We claim that in this case the equivalence relation \( {E}_{a} \) induced by the logic action is not Borel. Suppose not. Then \( {E}_{a} \in {... | It follows that \( W{O}^{\alpha } = \{ x \in {WO} : \left| x\right| \leq \alpha \} \in {\mathbf{\sum }}_{\beta }^{0} \) for every \( \alpha < {\omega }_{1} \) . Now take any Borel set \( A \) in \( {\mathbb{N}}^{\mathbb{N}} \) which is not of additive class \( \beta \) . Since \( {WO} \) is \( {\mathbf{\Pi }}_{1}^{1} \... | Yes |
Theorem 5.13.19 (Stern) Assume analytic determinacy. Let \( E \) be an analytic equivalence relation on a Polish space \( X \) such that all but countably many equivalence classes are of bounded Borel rank. Then the number of equivalence classes is \( \leq {\aleph }_{0} \) or perfectly many. | The proof this result is beyond the scope of this book. | No |
Theorem 5.14.1 (Kondô’s theorem) Let \( X, Y \) be Polish spaces. Every coanalytic set \( C \subseteq X \times Y \) admits a coanalytic uniformization. | We shall show that there is a sequence of coanalytic norms on a given co-analytic set with certain \ | No |
Corollary 5.14.5 Let \( X \) be a Polish space and \( A \subseteq X \) coanalytic. Then A admits a very good \( {\mathbf{\Pi }}_{1}^{1} \) -scale. | Proof. By 2.6.9 there is a closed set \( D \subseteq {\mathbb{N}}^{\mathbb{N}} \) and a continuous bijection \( f : D \rightarrow X \) . Now, \( {f}^{-1}\left( A\right) \cap D \) is a \( {\mathbf{\Pi }}_{1}^{1} \) subset of \( {\mathbb{N}}^{\mathbb{N}} \) and hence admits a very good \( {\mathbf{\Pi }}_{1}^{1} \) -scal... | Yes |
We consider the discretization of the boundary value problem for the ordinary differential equation\n\n\[ - {u}^{\prime \prime }\left( x\right) = f\left( {x, u\left( x\right) }\right) ,\;x \in \left\lbrack {0,1}\right\rbrack \]\n\n(2.1)\n\nwith boundary condition\n\n\[ u\left( 0\right) = u\left( 1\right) = 0. \]\n\n(2.... | For the approximate solution we choose an equidistant subdivision of the interval \( \left\lbrack {0,1}\right\rbrack \) by setting\n\n\[ {x}_{j} = {jh},\;j = 0,\ldots, n + 1, \]\n\nwhere the step size is given by \( h = 1/\left( {n + 1}\right) \) with \( n \in \mathbb{N} \). At the internal grid points \( {x}_{j}, j = ... | Yes |
Consider the linear integral equation\n\n\[ \varphi \left( x\right) - {\int }_{0}^{1}K\left( {x, y}\right) \varphi \left( y\right) {dy} = f\left( x\right) ,\;x \in \left\lbrack {0,1}\right\rbrack ,\] | For the numerical approximation we replace the integral by the rectangular sum\n\n\[ {\int }_{0}^{1}K\left( {x, y}\right) \varphi \left( y\right) {dy} \approx \frac{1}{n}\mathop{\sum }\limits_{{k = 1}}^{n}K\left( {x,{x}_{k}}\right) \varphi \left( {x}_{k}\right) \]\n\nwith equidistant grid points \( {x}_{k} = k/n, k = 1... | Yes |
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