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Proposition 4.3.14 No uncountable analytic \( A \subseteq \mathbb{R} \) has strong measure zero. | Null | No |
Theorem 4.3.17 Every coanalytic set is a union of \( {\aleph }_{1} \) Borel sets. | Proof. Let \( X \) be Polish and \( C \subseteq X \) coanalytic. By the Borel isomorphism theorem (3.3.13), without any loss of generality we may assume that \( X = {\mathbb{N}}^{\mathbb{N}} \) . By 4.1.20, there is a tree \( T \) on \( \mathbb{N} \times \mathbb{N} \) such that\n\n\[ \alpha \in C \Leftrightarrow T\left... | Yes |
Theorem 4.3.18 A coanalytic set is either countable or of cardinality \( {\aleph }_{1} \) or \( \mathfrak{c} \) . | Null | No |
Lemma 4.4.2 Suppose \( E = \mathop{\bigcup }\limits_{n}{E}_{n} \) cannot be separated from \( F = \mathop{\bigcup }\limits_{m}{F}_{m} \) by a Borel set. Then there exist \( m, n \) such that \( {E}_{n} \) cannot be separated from \( {F}_{m} \) by a Borel set. | Proof. Suppose for every \( m, n \) there is a Borel set \( {C}_{mn} \) such that\n\n\[ \n{E}_{n} \subseteq {C}_{mn}\text{ and }{F}_{m}\bigcap {C}_{mn} = \varnothing .\n\]\n\nIt is fairly easy to check that the Borel set\n\n\[ \nC = \mathop{\bigcup }\limits_{n}\mathop{\bigcap }\limits_{m}{C}_{mn}\n\]\nseparates \( E \)... | Yes |
Theorem 4.4.3 (Souslin) A subset A of a Polish space \( X \) is Borel if and only if it is both analytic and coanalytic; i.e., \( {\mathbf{\Delta }}_{1}^{1}\left( X\right) = {\mathcal{B}}_{X} \) . | Proof. The \ | No |
Proposition 4.4.4 Suppose \( {A}_{0},{A}_{1},\ldots \) are pairwise disjoint analytic subsets of a Polish space \( X \) . Then there exist pairwise disjoint Borel sets \( {B}_{0},{B}_{1},\ldots \) such that \( {B}_{n} \supseteq {A}_{n} \) for all \( n \) . | Proof. By 4.4.1, for each \( n \) there is a Borel set \( {C}_{n} \) such that\n\n\[ \n{A}_{n} \subseteq {C}_{n}\text{ and }{C}_{n} \cap \mathop{\bigcup }\limits_{{m \neq n}}{A}_{m} = \varnothing .\n\]\n\nTake\n\n\[ \n{B}_{n} = {C}_{n} \cap \mathop{\bigcap }\limits_{{m \neq n}}\left( {X \smallsetminus {C}_{m}}\right)\n... | Yes |
Theorem 4.4.5 Let \( E \subseteq X \times X \) be an analytic equivalence relation on a Polish space \( X \) . Suppose \( A \) and \( B \) are disjoint analytic subsets of \( X \) . Assume that \( B \) is invariant with respect to \( E \) (i.e., \( B \) is a union of \( E \) - equivalence classes). Then there is an \( ... | Proof. First we note the following. Let \( D \) be an analytic subset of \( X \) and \( {D}^{ * } \) the smallest invariant set containing \( D \) . Since\n\n\[ \n{D}^{ * } = {\pi }_{X}\left( {E\bigcap \left( {D \times X}\right) }\right) \n\]\n\nwhere \( {\pi }_{X} : X \times X \rightarrow X \) is the projection to the... | Yes |
Proposition 4.5.1 Let \( A \) be an analytic subset of a Polish space, \( Y \) a Polish space, and \( f : A \rightarrow Y \) a one-to-one Borel map. Then \( f : A \rightarrow \) \( f\left( A\right) \) is a Borel isomorphism. | Proof. Let \( B \subseteq A \) be Borel in \( A \) . We need to show that \( f\left( B\right) \) is Borel in \( f\left( A\right) \) . As both \( B \) and \( C = A \smallsetminus B \) are analytic and \( f \) Borel, \( f\left( B\right) \) and \( f\left( C\right) \) are analytic. Since \( f \) is one-to-one, these two se... | Yes |
Theorem 4.5.2 Let \( X, Y \) be Polish spaces, \( A \subseteq X \) analytic, and \( f : A \rightarrow Y \) any map. The following statements are equivalent\n\n(i) \( f \) is Borel measurable.\n\n(ii) \( \operatorname{graph}\left( f\right) \) is Borel in \( A \times Y \). \n\n(iii) \( \operatorname{graph}\left( f\right)... | Proof. We only need to show that (iii) implies (i). The other implications are quite easy to see. Let \( U \) be an open set in \( Y \). As\n\n\[ \n{f}^{-1}\left( U\right) = {\pi }_{X}\left( {\operatorname{graph}\left( f\right) \bigcap \left( {X \times U}\right) }\right) , \n\]\n\nwhere \( {\pi }_{X} : X \times Y \righ... | Yes |
Corollary 4.5.5 Let \( X \) be a standard Borel space and \( Y \) a metrizable space. Suppose there is a one-to-one Borel map \( f \) from \( X \) onto \( Y \) . Then \( Y \) is standard Borel and \( f \) a Borel isomorphism. | Proof. By 4.3.8, \( Y \) is separable. The result follows from 4.5.4. | Yes |
Theorem 4.5.7 (Blackwell - Mackey theorem, [13]) Let \( X \) be an analytic subset of a Polish space and \( \mathcal{A} \) a countably generated sub \( \sigma \) -algebra of the Borel \( \sigma \) -algebra \( {\mathcal{B}}_{X} \) . Let \( B \subseteq X \) be a Borel set that is a union of atoms of \( \mathcal{A} \) . T... | Proof. Let \( \left\{ {{B}_{n} : n \in \mathbb{N}}\right\} \) be a countable generator of \( \mathcal{A} \) . Consider the map \( f : X \rightarrow {2}^{\mathbb{N}} \) defined by\n\n\[ f\left( x\right) = \left( {{\chi }_{{B}_{0}}\left( x\right) ,{\chi }_{{B}_{1}}\left( x\right) ,\ldots }\right) ,\;x \in X. \]\n\nThen \... | Yes |
Corollary 4.5.10 Let \( X \) be an analytic subset of a Polish space and \( {\mathcal{A}}_{1} \) , \( {\mathcal{A}}_{2} \) two countably generated sub \( \sigma \) -algebras of \( {\mathcal{B}}_{X} \) with the same set of atoms. Then \( {\mathcal{A}}_{1} = {\mathcal{A}}_{2} \) . In particular, if \( \mathcal{A} \) is a... | Null | No |
Theorem 4.6.1 (The generalized first separation theorem, Novikov[90]) Let \( \\left( {A}_{n}\\right) \) be a sequence of analytic subsets of a Polish space \( X \) such that \( \\bigcap {A}_{n} = \\varnothing \) . Then there exist Borel sets \( {B}_{n} \\supseteq {A}_{n} \) such that \( \\bigcap {B}_{n} = \\varnothing ... | Proof of 4.6.1. (Mokobodzki [86]) Let \( \\left( {A}_{n}\\right) \) be a sequence of analytic sets that is not Borel separated and such that \( \\mathop{\\bigcap }\\limits_{n}{A}_{n} = \\varnothing \) . For each \( n \), fix a continuous surjection \( {f}_{n} : {\\mathbb{N}}^{\\mathbb{N}} \\rightarrow {A}_{n} \) . We g... | No |
Lemma 4.6.2 Let \( \left( {E}_{n}\right) \) be a sequence of subsets of \( X, k \in \mathbb{N} \), and \( {E}_{i} = \) \( \mathop{\bigcup }\limits_{n}{E}_{in} \) for \( i \leq k \) . Suppose \( \left( {E}_{n}\right) \) is not Borel separated. Then there exist \( {n}_{0},{n}_{1},\ldots ,{n}_{k} \) such that the sequence... | Proof. We prove the result by induction on \( k \) .\n\nInitial step: \( k = 0 \) . Suppose the result is not true. Hence, for every \( n \) , there is a sequence \( {\left( {B}_{in}\right) }_{i \in \mathbb{N}} \) of Borel sets such that\n\n(i) \( \mathop{\bigcap }\limits_{i}{B}_{in} = \varnothing \) ,\n\n(ii) \( {B}_{... | Yes |
Corollary 4.6.3 Let \( \\left( {A}_{n}\\right) \) be a sequence of analytic subsets of a Polish space \( X \) such that \( \\lim \\sup {A}_{n} = \\varnothing \) . Then there exist Borel sets \( {B}_{n} \\supseteq {A}_{n} \) such that \( \\lim \\sup {B}_{n} = \\varnothing \) . | Null | No |
Theorem 4.6.5 (Weak reduction principle for coanalytic sets) Let \( {C}_{0},{C}_{1},{C}_{2},\ldots \) be a sequence of coanalytic subsets of a Polish space such that \( \bigcup {C}_{n} \) is Borel. Then there exist pairwise disjoint Borel sets \( {B}_{n} \subseteq {C}_{n} \) such that \( \bigcup {B}_{n} = \bigcup {C}_{... | Proof. Let \( {A}_{n} = X \smallsetminus {C}_{n} \), where \( X = \mathop{\bigcup }\limits_{n}{C}_{n} \) . Then \( \left( {A}_{n}\right) \) is a sequence of analytic sets such that \( \mathop{\bigcap }\limits_{n}{A}_{n} = \varnothing \) . By 4.6.1, there exist Borel sets \( {D}_{n} \supseteq {A}_{n} \) such that \( \ma... | Yes |
Theorem 4.7.1 (Saint Raymond[97]) Let \( {A}_{0} \) and \( {A}_{1} \) be disjoint analytic subsets of \( X \times Y \) with the sections \( {\left( {A}_{0}\right) }_{x}, x \in X \), closed in \( Y \). Then there is a sequence \( \left( {B}_{n}\right) \) of Borel subsets of \( X \) such that \[ {A}_{1} \subseteq \mathop... | Proof. By 4.4.1, there is a Borel set containing \( {A}_{1} \) and disjoint from \( {A}_{0} \). So, without any loss of generality, we assume that \( {A}_{1} \) is Borel. For each \( n \), let \[ {C}_{n} = \left\{ {x \in X : {V}_{n} \subseteq {\left( {A}_{0}\right) }_{x}^{c}}\right\} \] Then \( {C}_{n} \) is coanalytic... | Yes |
Theorem 4.7.2 (Kunugui, Novikov) Suppose \( B \subseteq X \times Y \) is any Borel set with sections \( {B}_{x} \) open, \( x \in X \) . Then there is a sequence \( \left( {B}_{n}\right) \) of Borel subsets of \( X \) such that\n\n\[ B = \bigcup \left( {{B}_{n} \times {V}_{n}}\right) \] | Proof. Apply 4.7.1 to \( {A}_{0} = {B}^{c} \) and \( {A}_{1} = B \) . | No |
Corollary 4.7.3 Let \( {A}_{0} \) and \( {A}_{1} \) be disjoint analytic subsets of \( X \times Y \) with sections \( {\left( {A}_{0}\right) }_{x} \) and \( {\left( {A}_{1}\right) }_{x} \) closed for all \( x \in X \) . Then there exist disjoint Borel sets \( {B}_{0} \) and \( {B}_{1} \) with closed sections such that ... | Null | No |
Corollary 4.7.4 Suppose \( B \subseteq X \times Y \) is a Borel set with the sections \( {B}_{x} \) closed. Then there is a Polish topology \( \mathcal{T} \) finer than the given topology on \( X \) generating the same Borel \( \sigma \) -algebra such that \( B \) is closed relative to the product topology on \( X \tim... | Proof. By 4.7.2, write\n\n\[ \n{B}^{c} = \mathop{\bigcup }\limits_{n}\left( {{B}_{n} \times {V}_{n}}\right) \n\]\n\nthe \( {B}_{n} \) ’s Borel. By 3.2.5, take a finer Polish topology \( \mathcal{T} \) on \( X \) generating the same Borel \( \sigma \) -algebra such that \( {B}_{n} \) is \( \mathcal{T} \) -open. | Yes |
Example 4.7.9 (H. Sarbadhikari) Let \( A \subseteq \left\lbrack {0,1}\right\rbrack \) be an analytic non-Borel set and \( E \subseteq \left\lbrack {0,1}\right\rbrack \times {\mathbb{N}}^{\mathbb{N}} \) a closed set whose projection is \( A \) . Set \( B = \) \( E\bigcup \left( {\left( {\left\lbrack {0,1}\right\rbrack \... | Suppose not. Consider \( C = {F}^{-1}(\left( {0,1\rbrack }\right) \) . Then \( C \) is a Borel set with sections \( {C}_{x} \) open and whose projection is \( A \) . Hence \( A \) is Borel. (See the paragraph below.) We have arrived at a contradiction. | Yes |
Theorem 4.7.11 (Novikov) Let \( X \) and \( Y \) be Polish spaces and \( B \) a Borel subset of \( X \times Y \) with sections \( {B}_{x} \) compact. Then \( {\pi }_{X}\left( B\right) \) is Borel in \( X \) . | Proof. (Srivastava) Since every Polish space is homeomorphic to a \( {G}_{\delta } \) subset of the Hilbert cube \( \mathbb{H} \), without any loss of generality, we assume that \( Y \) is a compact metric space. Note that the sections \( {B}_{x} \) are closed in \( Y \) . By 4.7.4, there is a finer Polish topology on ... | Yes |
Corollary 4.7.12 Let \( X, Y \) be Polish spaces with \( {Y\sigma } \) -compact (equivalently, locally compact). Then the projection of every Borel set \( B \) in \( X \times Y \) with \( x \) -sections closed in \( Y \) is Borel. | Proof. Write \( Y = \mathop{\bigcup }\limits_{n}{Y}_{n},{Y}_{n} \) compact. Then\n\n\[ \n{\pi }_{X}\left( B\right) = \mathop{\bigcup }\limits_{n}{\pi }_{X}\left( {B\bigcap \left( {X \times {Y}_{n}}\right) }\right) \n\] \n\nNow apply 4.7.11. | Yes |
Theorem 4.8.1 Let \( \left( {G, \cdot }\right) \) be a Polish group and \( H \) a closed subgroup. Suppose \( E = \left\{ {\left( {x, y}\right) : x \cdot {y}^{-1} \in H}\right\} \) ; i.e., \( E \) is the equivalence relation induced by the right cosets. Then the \( \sigma \) -algebra of invariant Borel sets is countabl... | Proof. Let \( \left\{ {{U}_{n} : n \in \mathbb{N}}\right\} \) be a countable base for the topology of \( G \) . Put\n\n\[ \n{B}_{n} = \mathop{\bigcup }\limits_{{y \in H}}y \cdot {U}_{n} \n\]\n\nSo, the \( {B}_{n} \) ’s are Borel (in fact, open). We show that \( \left\{ {{B}_{n} : n \in \mathbb{N}}\right\} \) generates ... | Yes |
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} ... | Yes |
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... | Yes |
Example 4.8.13 Consider the additive group \( \left( {\mathbb{R}, + }\right) \) of real numbers. Let \( \left( {\mathbb{R},\mathcal{T}}\right) \) be the topological sum \( \left( {\mathbb{R}\smallsetminus \{ 0\} \text{, usual topology}}\right) \oplus \{ 0\} \) So, \( \mathcal{T} \) is generated by the usual open sets a... | Null | No |
Example 4.8.14 (G. Hjorth) Under AC, there is a discontinuous group isomorphism \( \varphi : \mathbb{R} \rightarrow \mathbb{R} \) . Take \( G \) to be \( \mathbb{R} \times \mathbb{R} \) with the product topology and the group operation defined by\n\n\[ \left( {r, s}\right) \cdot \left( {p, q}\right) = \left( {r + {2}^{... | Null | No |
Theorem 4.9.1 (Moschovakis) Every \( {\mathbf{\Pi }}_{1}^{1} \) set \( A \) in a Polish space \( X \) admits a \( {\mathbf{\Pi }}_{1}^{1} \) -norm \( \varphi : A \rightarrow {\omega }_{1} \) . | Null | 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 \( n > 1 \) . | No |
Example 4.9.17 Let \( {U}_{0},{U}_{1} \) be a universal pair of analytic subsets of \( {\mathbb{N}}^{\mathbb{N}} \times {\mathbb{N}}^{\mathbb{N}} \) (4.1.17). Suppose there exist pairwise disjoint analytic sets \( {V}_{0} \subseteq \) \( {U}_{0},{V}_{1} \subseteq {U}_{1} \) such that \( {V}_{0}\bigcup {V}_{1} = {U}_{0}... | Null | No |
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 4.9.21 Let \( {C}_{0} \) and \( {C}_{1} \) be disjoint coanalytic subsets of \( I = \left\lbrack {0,1}\right\rbrack \) that are not Borel separated; i.e., there is no Borel set containing \( {C}_{0} \) and disjoint from \( {C}_{1} \) . Let\n\n\[ \n{A}_{0} = \left( {I\times \{ 0\} }\right) \bigcup \left( {{C}_{0... | Null | No |
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 |
Example 4.10.2 (Separation capacity) Let \( X \) be a polish space. Define \( I : \mathcal{P}\left( {X \times X}\right) \rightarrow \{ 0,1\} \) by\n\n\[ I\left( A\right) = \left\{ \begin{array}{ll} 0 & \text{ if }{\pi }_{1}\left( A\right) \cap {\pi }_{2}\left( A\right) = \varnothing , \\ 1 & \text{ otherwise,} \end{arr... | Null | 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 |
Example 4.10.6 Consider \( I : \mathcal{P}\left( {\mathbb{N}}^{\mathbb{N}}\right) \rightarrow \{ 0,1\} \) defined by \[ I\left( A\right) = \left\{ \begin{array}{ll} 0 & \text{ if }A\text{ is contained in a }{K}_{\sigma }\text{ set,} \\ 1 & \text{ otherwise. } \end{array}\right. \] Then \( I \) satisfies the conditions ... | Null | No |
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... | Yes |
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 |
Corollary 4.12.2 Let \( X, Y \) be Polish spaces and \( B \subseteq X \times Y \) a Borel set. Then the set\n\n\[ Z = \\left\\{ {x \\in X : {B}_{x}\\text{ is a singleton }}\\right\\} \]\n\nis coanalytic. | Null | No |
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 |
Example 5.1.7 (Blackwell[16]) Let \( {C}_{1},{C}_{2} \) be two disjoint coanalytic subsets of \( \left\lbrack {0,1}\right\rbrack \) that cannot be separated by Borel sets. The existence of such sets has been shown in (4.9.17). Let \( {B}_{i} \) be a closed subset of \( \left\lbrack {0,1}\right\rbrack \times \sum \left(... | Null | No |
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... | No |
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. | Yes |
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). | Yes |
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 ... | 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 \( \mathcal{E} \) -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. | Yes |
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\... | Yes |
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\n\n\[ \mathcal{I}\left( x\right) = \{ I \subseteq Y : I\text{ is meager in }G\left( x\right) \} .\n\nBy imitati... | Null | 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 |
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