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Corollary 5.4.6. A subset \( E \) of \( \mathbb{Z} \) is a spectral set for \( M \) if and only if \( k\left( E\right) \) is contained in the closed linear span of all functions \( {e}_{n} \), where \( n \in \mathbb{Z} \smallsetminus E \) . | Proof. Because \( M \) is semisimple and regular, \( E \) is a spectral set if and only if \( k\left( E\right) = \overline{j\left( E\right) } \) . The statement now follows immediately from the preceding lemma. | No |
Corollary 5.4.7. Let \( E \subseteq \mathbb{Z} \) and \( m \in \mathbb{Z} \) . Then \( E \) is a spectral set for \( M \) if and only if \( E + m = \{ n + m : n \in E\} \) is a spectral set for \( M \) . | Proof. Of course, it suffices to show that if \( E \) is a spectral set, then so is \( E + m \) . Let \( f \in k\left( {E + m}\right) \) and let \( g \in M \) be defined by \( g\left( t\right) = f\left( t\right) {e}^{-{imt}} \) . Then, for each \( n \in E \) ,\n\n\[{\varphi }_{n}\left( g\right) = \frac{1}{2\pi }{\int }... | Yes |
Theorem 5.4.8. Let \( E \) and \( F \) be subsets of \( \mathbb{Z} \) such that \( E \subseteq F \) and \( F \smallsetminus E \) is finite. Then \( E \) is a set of synthesis for \( M \) if and only if \( F \) is a set of synthesis for \( M \) . | Proof. Suppose first that \( E \) is a spectral set. To show that \( F \) is a spectral set, proceeding inductively, it suffices to treat the case where \( F = E \cup \{ m\} \) for some \( m \in \mathbb{Z} \smallsetminus E \) . Let \( f \in k\left( F\right) \) and \( \epsilon > 0 \) be given. There exists \( g \in j\le... | Yes |
Theorem 5.4.9. Let \( E \) and \( F \) be subsets of \( \mathbb{Z} \) such that \( F \subseteq E \). (i) If \( F \) is a Ditkin set, so is \( E \). (ii) If \( E \) is a Ditkin set and \( E \smallsetminus F \) is finite, then \( F \) is a Ditkin set. | Proof. (i) Because, by Theorem 5.2.2, a closed countable union of Ditkin sets is a Ditkin set, it suffices to show that if \( F \) is a Ditkin set and \( m \in \mathbb{Z} \smallsetminus F \), then \( E = F \cup \{ m\} \) is a Ditkin set.\n\nLet \( f \in k\left( E\right) \) and \( \epsilon > 0 \) be given. Since \( F \)... | Yes |
Lemma 5.4.10. For a subset \( E \) of \( \mathbb{Z} \), the following conditions are equivalent.\n\n(i) \( E \) is a spectral set for \( M \) .\n\n(ii) \( \langle \mu, f\rangle = 0 \) for every \( f \in k\left( E\right) \) and every \( \mu \in {M}^{ * } \) with \( \sigma \left( \mu \right) \subseteq E \) . | Proof. (i) \( \Rightarrow \) (ii) Let \( f \in k\left( E\right) \) and \( \mu \in {M}^{ * } \) with \( \sigma \left( \mu \right) \subseteq E \) . Because \( E \) is a spectral set, by Lemma 5.4.5 \( k\left( E\right) \) is the closed linear span of functions \( {e}_{n}, n \in \mathbb{Z} \smallsetminus E \) . Thus there ... | Yes |
Proposition 5.4.11. For a subset \( E \) of \( \mathbb{Z} \) the following conditions are equivalent.\n\n(i) \( E \) is a spectral for \( M \) .\n\n(ii) If \( \mu \in {M}^{ * } \) is such that \( \sigma \left( \mu \right) \subseteq E \), then \( \mu \) is the \( {w}^{ * } \) -limit of functionals in the linear span of ... | Proof. (i) \( \Rightarrow \) (ii) Let \( \mu \in {M}^{ * } \) such that \( \sigma \left( \mu \right) \subseteq E \) . Because \( E \) is a spectral set, Lemma 5.4.10 implies that \( \langle \mu, f\rangle = 0 \) for every \( f \in k\left( E\right) \) . Towards a contradiction, assume that \( \mu \) does not belong to th... | Yes |
Theorem 5.4.12. Every subset of \( 4\mathbb{Z} \) is a spectral set for \( M \) . | Proof. Let \( E \) be a nonempty subset of \( 4\mathbb{Z} \) and \( \nu = {gdx} + \mu \in {M}^{ * } \) with \( \sigma \left( \nu \right) \subseteq E \) . Let \( p \) be any trigonometric polynomial and let \( q\left( t\right) = p\left( {t + \pi /2}\right) \) .\n\nThen\n\[ \widehat{q}\left( n\right) = {e}^{{in\pi }/2}{\... | Yes |
Lemma 5.4.13. Let \( f \in M \) be defined by\n\n\[ f\left( t\right) = \left\{ \begin{array}{rr} 1 & \text{ if }\left| t\right| \leq \pi /2 \\ - 1 & \text{ if }\pi /2 < \left| t\right| \leq \pi \end{array}\right. \]\n\nThen \( \widehat{f}\left( 0\right) = 0 \) and, for \( n \in \mathbb{Z} \smallsetminus \{ 0\} \) ,\n\n... | Proof. For \( n \in \mathbb{Z}, n \neq 0 \), we have\n\n\[ \widehat{f}\left( n\right) = \frac{1}{2\pi }\left( {-{\int }_{-\pi }^{-\pi /2}{e}^{-{int}}{dt} + {\int }_{-\pi /2}^{\pi /2}{e}^{-{int}}{dt} - {\int }_{\pi /2}^{\pi }{e}^{-{int}}{dt}}\right) \]\n\n\[ = \frac{1}{2\pi in}\left( {{e}^{{in\pi }/2} - {e}^{in\pi } - {... | Yes |
Lemma 5.4.14. Let \( \mu = {\delta }_{-\pi /2} + {\delta }_{\pi /2} \in {M}^{ * } \) . Then\n\n\[ \widehat{\mu }\left( n\right) = 2\cos \left( \frac{n\pi }{2}\right) \]\n\nfor every \( n \in \mathbb{Z} \) . In particular, \( \sigma \left( \mu \right) = 2\mathbb{Z} \) . | Proof. For each \( n \in \mathbb{Z} \) ,\n\n\[ \widehat{\mu }\left( n\right) = {\int }_{\mathbb{T}}{e}^{-{int}}{d\mu }\left( t\right) = {e}^{{in\pi }/2} + {e}^{-{in\pi }/2} = 2\cos \left( \frac{n\pi }{2}\right) . \]\n\nSince \( \cos \left( {{n\pi }/2}\right) \neq 0 \) if and only if \( n \in 2\mathbb{Z} \), we get that... | Yes |
Corollary 5.4.15. Let \( f \) and \( \mu \) be as in Lemmas 5.4.13 and 5.4.14, respectively. Then \( \langle \mu, f - p * f\rangle = 2 \) for every trigonometric polynomial \( p \) . | Proof. Notice first that \( \langle \mu, f\rangle = f\left( {-\pi /2}\right) + f\left( {\pi /2}\right) = 2 \) . Then\n\n\[ \langle \mu, f - p * f\rangle = 2 - \langle \mu, p * f\rangle = 2 - \mathop{\sum }\limits_{{n \in \mathbb{Z}}}\widehat{p}\left( n\right) \widehat{f}\left( n\right) \widehat{\mu }\left( n\right) \]\... | Yes |
Corollary 5.4.16. The union of two disjoint sets of synthesis for \( M \) need not be a set of synthesis. For example, \( 2\mathbb{Z} = 4\mathbb{Z} \cup \left( {4\mathbb{Z} + 2}\right) \) fails to be of synthesis even though both \( 4\mathbb{Z} \) and \( 4\mathbb{Z} + 2 \) are sets of synthesis. | Proof. By Theorem 5.4.12, \( 4\mathbb{Z} \) is a spectral set and hence so is \( 4\mathbb{Z} + 2 \) by Corollary 5.4.7. However, \( 2\mathbb{Z} \) is not of synthesis. Indeed, let \( f \) and \( \mu \) be as in Lemmas 5.4.13 and 5.4.14. Then \( f \in k\left( {2\mathbb{Z}}\right) \) and \( \sigma \left( \mu \right) = 2\... | Yes |
Theorem 5.4.17. Let \( E \) be a finite subset of \( \mathbb{Z} = \Delta \left( M\right) \) . Then \( E \) is a set of synthesis, but \( E \) fails to be a Ditkin set. | Proof. We first show that \( E \) is a set of synthesis. Since the trigonometric polynomials \( p \) are dense in \( M \) (Lemma 5.4.2) and \( \widehat{p} \) has finite support for any such \( p,\varnothing \) is a spectral set. Thus let \( E \) be nonempty and let \( f \in k\left( E\right) \) and \( \mu \in {M}^{ * } ... | Yes |
Theorem 5.5.1. Let \( G \) be a locally compact Abelian group.\n\n(i) \( {L}^{1}\left( G\right) \) satisfies Ditkin’s condition at infinity.\n\n(ii) Let \( K \) be a compact subset of \( \widehat{G} \) and \( \epsilon > 0 \) . Then there exists \( f \in {L}^{1}\left( G\right) \) such that \( \widehat{f} = 1 \) on \( K,... | Proof. (i) Because \( {L}^{1}\left( G\right) \) has an approximate identity, it suffices to show that \( {L}^{1}\left( G\right) \) is Tauberian. Let \( F \) denote the space of all functions \( f \in {L}^{2}\left( G\right) \) such that \( \widehat{f} \) equals almost everywhere a continuous function with compact suppor... | Yes |
Corollary 5.5.2. Every proper closed ideal of \( {L}^{1}\left( G\right) \) is contained in some maximal modular ideal. | Because \( \Delta \left( {{L}^{1}\left( G\right) }\right) = \widehat{G} \) is discrete when \( G \) is compact, Theorem 5.5.1 and Corollary 5.2.15 imply the following | No |
Corollary 5.5.4. Let \( f \in {L}^{1}\left( G\right) \) and let \( I \) denote the closed linear subspace of \( {L}^{1}\left( G\right) \) generated by all the translates \( {L}_{x}f, x \in G \) . Then the following conditions are equivalent.\n\n(i) \( I = {L}^{1}\left( G\right) \).\n\n(ii \( \widehat{f}\left( \alpha \r... | Proof. Suppose that \( I \neq {L}^{1}\left( G\right) \) . Then, by Proposition 1.4.7, \( I \) is a proper closed ideal of \( {L}^{1}\left( G\right) \) and hence, by Corollary 5.5.2, there exists a maximal modular ideal \( M \) of \( {L}^{1}\left( G\right) \) such that \( M \supseteq I \) . By Theorem 2.7.2, \( M \) is ... | Yes |
Lemma 5.5.6. Given \( f \in {L}^{1}\left( G\right) \) and \( \epsilon > 0 \), there exists \( g \in {\mathcal{F}}_{C}\left( G\right) \) such that \[ {\begin{Vmatrix}f * g - \widehat{f}\left( {1}_{G}\right) g\end{Vmatrix}}_{1} < \epsilon \] and hence also \[ \parallel f * g{\parallel }_{1} < C\left| {\widehat{f}\left( {... | Proof. For any \( g \in {L}^{1}\left( G\right) \), we have \[ {\begin{Vmatrix}f * g - \widehat{f}\left( {1}_{G}\right) g\end{Vmatrix}}_{1} \leq {\int }_{G}\left| {f\left( y\right) }\right| \cdot {\begin{Vmatrix}{L}_{y}g - g\end{Vmatrix}}_{1}{dy}. \] Choose a compact subset \( K \) of \( G \) such that \[ {\int }_{G \sm... | Yes |
Lemma 5.5.7. Let \( H \) be a closed subgroup of \( G \) and let \( f \in {L}^{1}\left( G\right) \) and \( \epsilon > 0 \) . Then there exists a measure \( \mu \in M\left( G\right) \) with the following properties.\n\n(i) \( \parallel \mu \parallel < 1 + \epsilon \) and \( \widehat{\mu } = 1 \) on a neighbourhood of \(... | Proof. We first establish the lemma for \( f \in {C}_{c}\left( G\right) \) . Choose \( C > 1 \) such that \( C < 1 + \epsilon \) and \( \left( {C - 1}\right) {\begin{Vmatrix}{T}_{H}f\end{Vmatrix}}_{1} < \epsilon /2 \) . By Lemma 5.5.6 there exists \( h \in {L}^{1}\left( H\right) \) such that \( \parallel h{\parallel }_... | Yes |
Theorem 5.5.8. Let \( G \) be a locally compact Abelian group, \( \Gamma \) a closed subgroup of \( \widehat{G} \), and \( \alpha \in \widehat{G} \). For every \( \delta > 0, k\left( {\alpha \Gamma }\right) \) possesses an approximate identity of norm bound \( 2 + \delta \) which is contained in \( j\left( {\alpha \Gam... | Proof. By the remark preceding the theorem, we can assume that \( \alpha = {1}_{G} \). The closed subgroup \( \Gamma \) of \( \widehat{G} \) is of the form \( \Gamma = \widehat{G/H} \), where \( H \) is the annihilator of \( \Gamma \) in \( \widehat{G} \); that is, \( H = \{ x \in G : \gamma \left( x\right) = 1 \) for ... | Yes |
Theorem 5.5.9. Let \( G \) be a locally compact Abelian group, \( H \) a closed subgroup of \( G \), and embed \( \widehat{G/H} \) into \( \widehat{G} \) . Let \( E \) be a closed subset of \( \widehat{G/H} \) . (i) \( E \) is a set of synthesis for \( {L}^{1}\left( {G/H}\right) \) if and only if \( E \) is a set of sy... | Proof. It follows from Theorem 5.2.6(i) that if \( E \) is a set of synthesis (respectively, Ditkin set) for \( {L}^{1}\left( G\right) \), then \( E \) is a set of synthesis (respectively, Ditkin set) for \( {L}^{1}\left( {G/H}\right) = {L}^{1}\left( G\right) /\ker {T}_{H} \) . Moreover, the converse conclusion of (i) ... | Yes |
Corollary 5.5.10. Let \( {\Gamma }_{1},\ldots ,{\Gamma }_{m} \) be closed subgroups of \( \widehat{G} \) and let \( {\gamma }_{1},\ldots ,{\gamma }_{m} \) be characters of \( G \) . For each \( j = 1,\ldots, m \), let \( {m}_{j} \in {\mathbb{N}}_{0} \) and for \( 1 \leq k \leq {m}_{j} \) , let \( {\Delta }_{jk} \) be a... | Proof. Finite unions of Ditkin sets are Ditkin sets (Lemma 5.2.1). Therefore we can assume that \( m = 1 \) . Moreover, if \( F \) is a Ditkin set then so is \( {\gamma F} \) for every \( \gamma \in \widehat{G} \) . Thus we can further assume that \( E \) is of the form \( E = \) \( \Gamma \smallsetminus \mathop{\bigcu... | Yes |
Corollary 5.6.2. Let \( G \) be a compact Abelian group and \( I \) a closed ideal of \( {L}^{1}\left( G\right) \) . Then \( I \) has a bounded approximate identity if and only if \( I = \mu * {L}^{1}\left( G\right) \) for some idempotent measure \( \mu \in M\left( G\right) \) . | Proof. Let \( I \) have a bounded approximate identity. Then, by Proposition 5.6.1, there exists an idempotent measure \( \mu \) on \( b\left( G\right) = G \) such that \( \widehat{\mu } \) equals the characteristic function of \( \widehat{G} \smallsetminus h\left( I\right) \) . Now,\n\n\[ h\left( {\mu * {L}^{1}\left( ... | Yes |
Proposition 5.6.3. If \( \mu \in F\left( G\right) \), then the support group of \( \mu \), the smallest closed subgroup of \( G \) whose complement is a \( \mu \) -null set, is compact. | Proof. Assume that \( \mu \neq 0 \) and let \( H \) be the support group of \( \mu \) . Then \( \mu \) can be regarded as an element of \( F\left( H\right) \) . We show that \( \widehat{H} \) is discrete and hence \( H \) is compact.\n\nLet \( \gamma \) be any nontrivial character of \( H \) . We claim that \( \gamma \... | Yes |
Proposition 5.6.8. Let \( G \) be a compact Abelian group and \( \mu \in F\left( G\right) \) . Then the set\n\n\[ S\left( \mu \right) = \{ \alpha \in \widehat{G} : \widehat{\mu }\left( \alpha \right) \neq 0\} \]\n\nbelongs to \( \mathcal{R}\left( \widehat{G}\right) \) . | Proof. For \( k \in {\mathbb{Z}}^{ * } = \mathbb{Z} \smallsetminus \{ 0\} \), let \( S{\left( \mu \right) }_{k} = \{ \alpha \in \widehat{G} : \widehat{\mu }\left( \alpha \right) = k\} \) . Since the range of \( \widehat{\mu } \) is finite, it suffices to show that \( S{\left( \mu \right) }_{k} \in \mathcal{R}\left( \wi... | Yes |
Lemma 5.7.1. Let \( E \) and \( F \) be closed subsets of \( \Delta \left( A\right) \) and \( \Delta \left( B\right) \), respectively, and let \( \psi \in F \) . Then\n\n\[{\phi }_{\psi }\left( {j\left( {E \times F}\right) }\right) \subseteq j\left( E\right) . | Proof. Let \( x \in j\left( {E \times F}\right) \) . There exists a compact subset \( C \) of \( \Delta \left( {A\widehat{ \otimes }B}\right) = \) \( \Delta \left( A\right) \times \Delta \left( B\right) \) such that \( \widehat{x} = 0 \) outside of \( C \) and \( C \cap \left( {E \times F}\right) = \varnothing \) . Now... | Yes |
Theorem 5.7.2. Let \( A \) and \( B \) be regular commutative Banach algebras and suppose that \( A{\widehat{ \otimes }}_{\pi }B \) is semisimple. Let \( E \) and \( F \) be closed subsets of \( \Delta \left( A\right) \) and \( \Delta \left( B\right) \), respectively. If \( E \times F \) is a set of synthesis (Ditkin s... | Proof. Let \( a \in A \) and \( \epsilon > 0 \) be given. Choose \( \psi \in F \) and \( b \in B \) such that \( \psi \left( b\right) = 1 \) . If \( E \times F \) is a set of synthesis, there exists \( x \in j\left( {E \times F}\right) \) such that \( \parallel x - a \otimes b\parallel \leq \epsilon \) . Then \( {\phi ... | Yes |
Corollary 5.7.4. Let \( G \) be a compact Abelian group and \( A \) a semisimple and regular commutative Banach algebra. Suppose that \( A \) is Tauberian and has a bounded approximate identity. Then spectral synthesis holds for \( {L}^{1}\left( {G, A}\right) \) if and only if it holds for \( A \) . | Proof. Recall first that \( {L}^{1}\left( {G, A}\right) = {L}^{1}\left( G\right) {\widehat{ \otimes }}_{\pi }A \) is semisimple by Theorem 2.11.6 because \( {L}^{1} \) -spaces share the approximation property. Thus, if spectral synthesis holds for \( {L}^{1}\left( {G, A}\right) \) then it holds for \( A \) (Theorem 5.7... | Yes |
Lemma 5.7.5. Suppose that \( A \) and \( B \) are Tauberian commutative Banach algebras, and let \( E \) and \( F \) be closed subsets of \( \Delta \left( A\right) \) and \( \Delta \left( B\right) \), respectively. Then\n\n\[ j\left( E\right) \otimes B + A \otimes j\left( F\right) \subseteq \overline{j\left( {E \times ... | Proof. It suffices to show that if\n\n\[ x = \mathop{\sum }\limits_{{k = 1}}^{n}{a}_{k} \otimes {b}_{k} + \mathop{\sum }\limits_{{l = 1}}^{m}{c}_{l} \otimes {d}_{l} \]\n\nwhere \( {a}_{k} \in j\left( E\right) ,{b}_{k} \in B,{c}_{l} \in A \), and \( {d}_{l} \in j\left( F\right) \), then \( x \in \overline{j\left( {E \ti... | Yes |
Lemma 5.7.6. Let \( F \) be a closed subset of \( \Delta \left( A\right) \) and \( \varphi \in \Delta \left( A\right) \). (i) Every \( x \in k\left( {\{ \varphi \} \times F}\right) \) has a representation of the form \[ x = \mathop{\sum }\limits_{{j = 1}}^{\infty }\left( {{a}_{j} \otimes {b}_{j}}\right) + e \otimes b \... | Proof. (i) Let \( x \in k\left( {\{ \varphi \} \times F}\right) \). Then \( \mathop{\sum }\limits_{{j = 1}}^{\infty }{y}_{j} \otimes {b}_{j} \), where \( {y}_{j} \in A,{b}_{j} \in B \), and \( \mathop{\sum }\limits_{{j = 1}}^{\infty }\begin{Vmatrix}{y}_{j}\end{Vmatrix} \cdot \begin{Vmatrix}{b}_{j}\end{Vmatrix} < \infty... | Yes |
Corollary 5.7.7. Let \( A \) and \( B \) be regular and Tauberian commutative Banach algebras and suppose that \( A{\widehat{ \otimes }}_{\pi }B \) is semisimple. Let \( \{ \varphi \} \) be a spectral set for \( A \) and \( F \subseteq \Delta \left( B\right) \) a spectral set for \( B \) . Then \( \{ \varphi \} \times ... | Proof. It follows from part (ii) of Lemma 5.7.6 and the hypotheses that\n\n\[ k\left( {\{ \varphi \} \times F}\right) = k\left( \varphi \right) {\widehat{ \otimes }}_{\pi }B + A \otimes k\left( F\right) \]\n\n\[ = \overline{j\left( \varphi \right) }{\widehat{ \otimes }}_{\pi }B + A \otimes \overline{j\left( F\right) } ... | Yes |
Theorem 5.7.8. Let \( A \) and \( B \) be unital regular commutative Banach algebras. Suppose that \( A{\widehat{ \otimes }}_{\pi }B \) is semisimple, and let \( E \) be a closed subset of \( \Delta \left( {A{\widehat{ \otimes }}_{\pi }B}\right) \) such that the set\n\n\[ \n\{ \varphi \in \Delta \left( A\right) : \left... | Proof. We apply Theorem 5.2.13 to \( A{\widehat{ \otimes }}_{\pi }B, T = \Delta \left( A\right) \) and the projection\n\n\[ \n\phi : \Delta \left( {A{\widehat{ \otimes }}_{\pi }B}\right) = \Delta \left( A\right) \times \Delta \left( B\right) \rightarrow \Delta \left( A\right) ,\;\left( {\varphi ,\psi }\right) \rightarr... | Yes |
Corollary 5.7.9. Let \( X \) be a compact Hausdorff space and \( A \) a regular and semisimple commutative Banach algebra with identity. Suppose that \( \Delta \left( A\right) \) is scattered and that every singleton \( \{ \varphi \} ,\varphi \in \Delta \left( A\right) \), is a spectral set and satisfies condition (D).... | Proof. We only have to note that \( A{\widehat{ \otimes }}_{\pi }C\left( X\right) \) is semisimple because \( A \) and \( C\left( X\right) \) are semisimple and \( C\left( X\right) \) has the approximation property. | Yes |
(ii) If \( A = \mathop{\bigcup }\limits_{n}{A}_{n} \), then\n\n\[{m}^{ * }\left( A\right) \leq \mathop{\sum }\limits_{n}{m}^{ * }\left( {A}_{n}\right)\] | Proof. (ii) We can, and do, assume that \( \mathop{\sum }\limits_{n}{m}^{ * }\left( {A}_{n}\right) < \infty \) . With this in mind, let \( \epsilon > 0 \) be given, and choose for each \( n \) a sequence \( \left( {I}_{n, j}\right) \) of intervals that cover \( {A}_{n} \) and satisfy\n\n\[ \mathop{\sum }\limits_{j}\ope... | Yes |
(i) If \( {F}_{1} \) and \( {F}_{2} \) are disjoint closed bounded sets, then\n\n\[ \n{m}^{ * }\left( {{F}_{1} \cup {F}_{2}}\right) = {m}^{ * }\left( {F}_{1}\right) + {m}^{ * }\left( {F}_{2}\right) \n\] | Proof. (i) Let \( \delta > 0 \) be chosen so that no interval of diameter less than \( \delta \) meets both \( {F}_{1} \) and \( {F}_{2} \) (e.g., \( \delta < \frac{1}{2}d\left( {{F}_{1},{F}_{2}}\right) \), where \( d\left( {{F}_{1},{F}_{2}}\right) \) is the distance between the disjoint closed sets \( {F}_{1} \) and \... | Yes |
(i) If \( A \) and \( B \) are measurable, then so is \( A \cap B \) . | Proof. (i) Pick \( {F}_{A},{F}_{B} \) closed and \( {G}_{A},{G}_{B} \) open such that\n\n\[ {F}_{A} \subseteq A \subseteq {G}_{A}\text{ and }{m}^{ * }\left( {{G}_{A} \smallsetminus {F}_{A}}\right) \leq \frac{\epsilon }{2} \]\n\n\[ {F}_{B} \subseteq B \subseteq {G}_{B}\text{ and }{m}^{ * }\left( {{G}_{B} \smallsetminus ... | Yes |
Theorem 1.7 (The Brunn-Minkowski Theorem). Let \( n \geq 1 \), and let \( {\lambda }_{n} \) denote Lebesgue measure on \( {\mathbb{R}}^{n} \). If \( A, B \), and \( A + B \) are measurable subsets of \( {\mathbb{R}}^{n} \), then\n\n\[{\left( {\lambda }_{n}\left( A + B\right) \right) }^{1/n} \geq {\left( {\lambda }_{n}\... | Proof. To prove (BM), suppose that each of \( A \) and \( B \) is the union of finitely many rectangles whose interiors are disjoint. We proceed by induction on the total number of rectangles in \( A \) and \( B \). It is important to realize that the inequality is unaffected if we translate \( A \) and \( B \) indepen... | Yes |
Theorem 1.8. For any Borel set \( B \subseteq {\mathbb{R}}^{n} \) we have\n\n\[ \n{\lambda }_{n}\left( B\right) \leq {\left( \frac{\operatorname{diam}B}{2}\right) }^{n}{\lambda }_{n}\left( {B}_{{\ell }_{n}^{2}}\right) \n\]\n\nwhere as usual \( {B}_{{\ell }_{n}^{2}} \) denotes the closed unit ball of \( {\mathbb{R}}^{n}... | Proof. Without loss of sleep we can assume that\n\n\[ \nd \mathrel{\text{:=}} \operatorname{diam}B < \infty \text{.} \n\]\n\nThis in mind, realize that if \( x, y, \in B \), then \( \parallel x - y\parallel \leq d \) . Hence\n\n\[ \nB - B \subseteq d{B}_{{\ell }_{n}^{2}} \n\]\n\nand so\n\n\[ \n2{\lambda }_{n}{\left( B\... | Yes |
Theorem 1.9 (Vitali). Let \( \mathcal{F} \) be a family of closed balls in \( {\mathbb{R}}^{n} \) that covers a set \( E \) in the sense of Vitali, that is, given an \( \epsilon > 0 \) and \( x \in E \), there is \( {aB} \in \mathcal{F} \) such that the diameter of \( B \) is less than \( \epsilon \) and \( x \in B \) ... | Proof. First we suppose \( E \) is bounded and contained in the bounded open set \( G \) . We disregard any members of \( \mathcal{F} \) that aren’t contained in \( G \) as well as those that don’t intersect \( E \) . The resulting family, which we will still refer to as \( \mathcal{F} \) covers \( E \) in the sense of... | No |
Theorem 1.10. Any uncountable \( {\mathcal{G}}_{\delta } \) subset of \( \mathbb{R} \) contains a homeomorphic copy of the Cantor set \( \Delta \) . | Proof. Let \( E \) be such a set. Then there exists a descending sequence \( \left( {G}_{n}\right) \) of open sets in \( \mathbb{R} \) so that \( E = \mathop{\bigcap }\limits_{n}{G}_{n} \) . Let \( F \) be the set of all \( x \in E \) for which given any open set \( U \) containing \( x, U \cap E \) is uncountable. Not... | Yes |
Theorem 1.11 (F. Bernstein). There exists a set \( B \) of real numbers such that both \( B \) and its complement \( {B}^{c} \) meet every uncountable closed set in \( \mathbb{R} \) . | Proof. We call on the well-ordering principle. Well-order \( \mathbb{R} \) . Using facts established above, we can also well-order the collection \( \mathcal{F} \) of all uncountable closed subsets of \( \mathbb{R} \) (each of which, is, by the way, of cardinality \( c \) ); we can index this well-ordering by the ordin... | Yes |
Theorem 1.12 (Isoperimetric Inequality). Among convex bodies of a given volume, Euclidean balls have the least surface area. | Proof. Let \( C \) be a convex, compact set in \( {\mathbb{R}}^{n} \) whose \( n \) -dimensional volume is that of\n\n\[ B \mathrel{\text{:=}} {B}_{{l}_{n}^{2}}.\]\n\nThe surface area of \( C \) can be expressed by\n\n(1)\n\n\[ \operatorname{sa}\left( {\partial C}\right) = \mathop{\lim }\limits_{{\epsilon \searrow 0}}\... | Yes |
Theorem 2.2 (Carathéodory). Let \( \mu \) be an outer measure on the set \( \Omega \) . Then\n\n(i) if \( \mu \left( N\right) = 0 \), then \( N \) is \( \mu \) -measurable;\n\n(ii) \( E \) is \( \mu \) -measurable if and only if \( {E}^{c} \) is;\n\n(iii) if \( \left( {E}_{n}\right) \) is a sequence of \( \mu \) -measu... | Proof. (i) Suppose \( A \subseteq N \) and \( B \subseteq {N}^{c} \) . Of course \( \mu \left( A\right) = 0 \) . Hence\n\n\[ \mu \left( B\right) \leq \mu \left( {A \cup B}\right) \leq \mu \left( A\right) + \mu \left( B\right) = \mu \left( B\right) .\n\nSqueezy says all are equal and so \( N \) is \( \mu \) -measurable.... | Yes |
Theorem 2.4. If \( \tau \) is a premeasure defined on a class \( \mathcal{C} \) of subsets of the space \( \Omega \), then the set function \( \mu \)\n\n\[ \mu \left( E\right) \mathrel{\text{:=}} \inf \left\{ {\mathop{\sum }\limits_{i}\tau \left( {C}_{i}\right) : {C}_{i} \in \mathcal{C}, E \subseteq \mathop{\bigcup }\l... | Proof. It is plain and easy to see that the set function \( \mu \) as defined from \( \tau \) satisfies properties (i)-(iii) of the definition of a measure. Only \ | No |
Theorem 2.5. \( \mu \) constructed by Method II is an outer measure. | Proof. The only possible stumbling point to this is the countable subadditivity, so let’s see why \( \mu \) is countably subadditive. To this end, let \( \left( {E}_{n}\right) \) be a sequence of subsets of \( \Omega \) and consider the quantities\n\n\[ \mu \left( {\mathop{\bigcup }\limits_{n}{E}_{n}}\right) \text{ and... | Yes |
Theorem 2.8. If \( \mu \) is a metric outer measure on the metric space \( \Omega \), then every closed subset of \( \Omega \) is \( \mu \) -measurable. Consequently if \( \mu \) is a metric outer measure on the metric space \( \Omega \) , then every Borel subset of \( \Omega \) is \( \mu \) -measurable. | Proof. Suppose \( F \) is a closed subset of the metric space \( \Omega \) with metric \( \rho \) , and let \( A \subseteq F \) and \( B \subseteq {F}^{c} \) be nonempty sets. For each \( n \) let\n\n\[ \n{B}_{n} \mathrel{\text{:=}} \left\{ {x \in B : \mathop{\inf }\limits_{{y \in F}}\rho \left( {x, y}\right) > \frac{1... | Yes |
Theorem 2.12. Let\n\n\[ \n{C}_{0} = \{ x = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \subseteq {\mathbb{R}}^{n} : 0 \leq {x}_{i} < 1, i = 1,2,\ldots, n\} \n\] \n\nand \n\n\[ \n{\kappa }_{n} = {\mu }^{\left( n\right) }\left( {C}_{0}\right) \n\] \n\nThen \n\n\[ \n{\mu }^{\left( n\right) }\left( E\right) = {\kappa }_{n}{\la... | Proof. By definition, \n\n\[ \n{\mu }^{\left( n\right) }\left( {C}_{0}\right) = {\kappa }_{n}{\lambda }_{n}\left( {C}_{0}\right) \n\] \n\nSince both \( {\mu }^{\left( n\right) } \) and \( {\lambda }_{n} \) are invariant under translations and are homogeneous of order \( n \) (i.e., for any \( t \geq 0,{\mu }^{\left( n\... | Yes |
Theorem 2.13. \[ {\kappa }_{n} = {\left( \frac{4}{\pi }\right) }^{n/2}\Gamma \left( \frac{n + 2}{2}\right) \] | Proof. To compute \( {\kappa }_{n} \) we need to know the volume, \( {\lambda }_{n}\left( {B}_{{\ell }_{2}^{n}}\right) \), of the closed unit ball in \( {\mathbb{R}}^{n} \) . Cutting to the quick, this is given by \[ {\lambda }_{n}\left( {B}_{{\ell }_{2}^{n}}\right) = \frac{{\pi }^{n/2}}{\Gamma \left( \frac{n + 2}{2}\r... | Yes |
Theorem 3.1. Let \( G \) be a topological group. If \( U \) is an open set containing the identity \( e \), then there is an open set \( V \) containing \( e \) such that \( e \in V \subseteq \) \( \overline{V} \subseteq U. \) Consequently, a \( {T}_{0} \) topological group is regular and so Hausdorff. | Proof. First things first: Let \( U \) be an open set that contains the identity \( e \) . By continuity of multiplication, there is an open set \( W \) containing \( e \) such that \( W \cdot W \subseteq U \) . If we set \( V = W \cap {W}^{-1} \), then we have an open set that contains \( e \), is symmetric \( \left( ... | Yes |
Proposition 3.2. Every open subgroup of a topological group is closed. | Proof. Let \( H \) be an open subgroup of the topological group \( G \) . Take \( g \in \) \( \bar{H} \) . Every open set that contains \( g \) intersects \( H;{gH} \) is such an open set.\n\nTherefore \( {gH} \cap H \neq \varnothing \) . Since cosets are either the same or disjoint, \( {gH} = H \) . Thus\n\n\[ g = {ge... | Yes |
Corollary 3.7. \( {GL}\left( {n,\mathbb{C}}\right) \) is a locally compact metrizable topological group. | Proof. After all, \[ {GL}\left( {n,\mathbb{C}}\right) = \mathop{\det }\limits^{ \leftarrow }\left( {\{ z \in \mathbb{C} : z \neq 0\} }\right) \] and so \( {GL}\left( {n,\mathbb{C}}\right) \) is homeomorphic to an open subset of a locally compact metric space, \( {\mathbb{C}}^{{n}^{2}} \) . Our comments about continuity... | Yes |
Theorem 3.8. The groups \( U\left( n\right), O\left( n\right) ,{SU}\left( n\right) \) , and \( {SO}\left( n\right) \) are compact metric topological groups. | Proof. Each of \( O\left( n\right) ,{SU}\left( n\right) ,{SO}\left( n\right) \) are closed subgroups of \( U\left( n\right) \), so it’s enough to establish that \( U\left( n\right) \) is compact. Now \( \alpha \in U\left( n\right) \) precisely when \( {\alpha }^{t}\bar{\alpha } = {\operatorname{id}}_{{\mathbb{C}}^{n}} ... | Yes |
Corollary 3.10. Let \( G \) be a topological group. If \( G \) has a countable neighborhood base at \( \{ e\} \), then \( G \) is metrizable. In this case the metric can be taken to be left invariant. | Proof. Suppose \( \left\{ {{V}_{n} : n \in \mathbb{N}}\right\} \) is a countable open base at \( e \) . Let \( {U}_{1} = \) \( {V}_{1} \cap {V}_{1}^{-1} \), and let \( {U}_{2} \) be a symmetric open neighborhood of \( e \) such that \( {U}_{2} \subseteq {U}_{1} \cap {V}_{2} \) and \( {U}_{2} \cdot {U}_{2} \subseteq {U}... | Yes |
Corollary 3.11. Let \( G \) be a topological group, let \( a \in G \), and let \( F \) be a closed subset of \( G \) such that \( a \notin F \) . Then there is a continuous real function \( \chi \) on \( G \) such that \( \chi \left( a\right) = 0 \) and \( \chi \left( x\right) = 1 \) for all \( x \in F \) . Consequentl... | Proof. Let \( {U}_{1} \) be a symmetric neighborhood of \( e \) such that \( \left( {a{U}_{1}}\right) \cap F = \varnothing \) . Choose a sequence \( \left( {{U}_{n} : n \geq 2}\right) \) of open neighborhoods of \( e \) such that each \( {U}_{n} \) is symmetric, \( {U}_{n + 1} \cdot {U}_{n + 1} \subseteq {U}_{n} \), an... | Yes |
Theorem 3.12. Any locally compact topological group is paracompact, hence normal. | Proof. Let \( G \) be a locally compact topological group, and let \( V \) be an open set in \( G \) containing the identity of \( G \) and having a compact closure \( \bar{V} \) . By Theorem 3.1 there exists a symmetric open set \( U \) such that\n\n\[ e \in U \subseteq \bar{U} \subseteq U \cdot U \subseteq V. \]\n\nL... | Yes |
Theorem 3.16. The product of Hausdorff spaces is a Hausdorff space. | Proof. If \( x, y \) are distinct members of \( S = \mathop{\prod }\limits_{\mathcal{A}}{S}_{\alpha } \), then there is an \( \alpha \in \mathcal{A} \) so that \( {x}_{\alpha } \neq {y}_{\alpha } \) ; since \( {S}_{\alpha } \) is a Hausdorff space there are open sets \( U \) and \( \begin{matrix} V & & \text{in}\;{S}_{... | No |
Theorem 3.17 (Tychonoff’s Product Theorem). The product of compact spaces is compact. | Proof. (Chevalley and Frink [17]). Let \( \mathcal{F} \) be a family of subsets of the product \( S \) of the compact spaces \( {S}_{\alpha },\alpha \in \mathcal{A} \), and suppose that any finite subfamily of \( \mathcal{F} \) has a nonempty intersection. Our aim is to show\n\n\[ \mathop{\bigcap }\limits_{F}F \neq \va... | Yes |
Theorem 4.1 (Hahn-Banach Theorem). Let \( X \) be a real linear space, and let \( S \) be a linear subspace of \( X \) . Suppose that \( p : X \rightarrow \mathbb{R} \) is a subadditive, positively homogeneous functional and \( f : S \rightarrow \mathbb{R} \) is a linear functional with \( f\left( s\right) \leq p\left(... | Proof. Our first task is to see how to extend a functional like \( f \) one dimension at a time while preserving the domination by \( p \) . With this in mind, let \( x \in X \smallsetminus S \), and notice that for any linear combination \( s + {\alpha x} \) of a vector in \( S \) and \( x \), whatever the linear exte... | No |
Proposition 4.4. If \( G \) is a compact metrizable group, then \( A \cong B \) whenever \( A = {gB} \) for some \( g \in G \) is a congruence. | Proof. Let \( \left( {S}_{n}\right) \) be a sequence of concentric open balls (with common center \( c) \) with radii tending to zero. Suppose \( \left( {A}_{n}\right) \) satisfies \( {A}_{n} \cong {S}_{n} \) for each \( n \) and suppose \( {a}_{n},{b}_{n} \in {A}_{n} \) have \( \mathop{\lim }\limits_{n}{a}_{n} = a,\ma... | Yes |
Proposition 4.5. Let \( Q \) be a compact metric space, and let \( G \) be a transitive equicontinuous group of homeomorphisms of \( Q \) onto itself. If \( A, B \in Q \), then \( A \cong B \) whenever \( A = g\left( B\right) \) for some \( g \in G \) is a congruence. | Proof. Transitivity of \( G \) is simply the feature that if \( {q}_{1},{q}_{2} \in Q \), then there is a \( g \in G \) so that \( g\left( {q}_{1}\right) = {q}_{2} \) . Because members of \( G \) are homeomorphisms of \( Q \) onto \( Q \) ,(iv) and (v) of Definition 4.3 are so. Transitivity of \( G \) assures us of (vi... | Yes |
Proposition 4.6. Suppose \( A, B \), and \( C \) are nonempty open subsets of \( Q \) . Then\n\n(v) If \( d\left( {A, B}\right) = \) distance from \( A \) to \( B \) is positive (so \( \bar{A} \cap \bar{B} = \varnothing \) ) and \( \left( {S}_{n}\right) \) is sequence of open concentric balls with radii tending to zero... | Proof. Item (v) requires some serious and careful attention. Suppose (v) fails. Then there is \( \left( {n}_{k}\right) \) so that\n\n\[ h\left( {A \cup B,{S}_{{n}_{k}}}\right) < h\left( {A,{S}_{{n}_{k}}}\right) + h\left( {B,{S}_{{n}_{k}}}\right) \n\nfor each \( k \) . We can plainly suppose that the \( {n}_{k} \) ’s ar... | Yes |
Proposition 4.7. If \( A \) and \( B \) are open sets, then\n\n(i) \( 0 < l\left( A\right) < \infty \), as long as \( A \neq \varnothing \).\n\n(ii) \( A \subseteq B \Rightarrow l\left( A\right) \leq l\left( B\right) \).\n\n(iii) \( l\left( {A \cup B}\right) \leq l\left( A\right) + l\left( B\right) \).\n\n(iv) \( A \co... | Proof. To see Proposition 4.7(v), note that by Proposition 4.6(v), there exists an \( N \) such that for all \( n \geq N \),\n\n\[ h\left( {A \cup B,{S}_{n}}\right) = h\left( {A,{S}_{n}}\right) + h\left( {B,{S}_{n}}\right) .\n\]\n\nFrom this we easily see that for all \( n \geq N \),\n\n\[ {l}_{n}\left( {A \cup B}\righ... | No |
Theorem 4.8. Let \( Q \) be a compact metric space. Then\n\n(i) \( 0 \leq \lambda \left( X\right) \) .\n\n(ii) If \( X \) is a nonempty open subset of \( Q \), then \( 0 < \lambda \left( X\right) < \infty \) .\n\n(iii) \( X \subseteq Y \subseteq Q \Rightarrow \lambda \left( X\right) \leq \lambda \left( Y\right) \) .\n\... | Proof. Items (i)-(iv) tell us that \( \lambda \) is an outer measure, item (v) assures us that \( \lambda \) respects congruence, and item (vi) says that \( \lambda \) is a \ | No |
Lemma 4.11. For any \( x \in \mathcal{C},\bar{\int }\left( x\right) = f\left( x\right) \) . | Proof. On the one hand, we can let \( {x}_{n} = x \) for all \( n \) and \( m = x \) . This done, we plainly have\n\n\[ \mathop{\liminf }\limits_{n}{x}_{n} \geq x,\text{ and }{x}_{n} \geq m \]\n\nhence\n\n\[ \bar{\int }\left( x\right) \leq \mathop{\liminf }\limits_{n}f\left( {x}_{n}\right) = f\left( x\right) \]\n\nOn t... | Yes |
Lemma 4.12. If \( {z}_{1},{z}_{2} \in {\mathcal{L}}^{ * } \) with \( \bar{\int }\left( {z}_{1}\right) ,\bar{\int }\left( {z}_{2}\right) < \infty \), then \( \bar{\int }\left( {{z}_{1} + {z}_{2}}\right) \leq \) \( \bar{\int }\left( {z}_{1}\right) + \bar{\int }\left( {z}_{2}\right) . | Proof. Suppose \( {P}_{1},{P}_{2} \) are numbers such that\n\n\[ \int \left( {z}_{1}\right) < {P}_{1}\text{ and }\int \left( {z}_{2}\right) < {P}_{2} \]\n\nThere are sequences \( \left( {x}_{n}^{\left( 1\right) }\right) ,\left( {x}_{n}^{\left( 2\right) }\right) \subseteq \mathcal{C} \) and functions \( {m}_{1},{m}_{2} ... | Yes |
Lemma 4.13. For any \( z \in {\mathcal{L}}^{ * } \) , \[ \int \left( z\right) \leq \int \left( z\right) \] | Proof. There is nothing to prove if \( \bar{\int }\left( z\right) = + \infty \) . If \( \bar{\int }\left( {-z}\right) = + \infty \), then \( \int \left( z\right) = - \bar{\int }\left( {-z}\right) \), so again there is nothing to prove. If \( \bar{\int }\left( z\right) ,\bar{\int }\left( {-z}\right) < \infty \) , then L... | Yes |
Lemma 4.14. If \( z \in {\mathcal{L}}^{ * } \) and \( \bar{f}\left( z\right) < \infty \), then\n\n\[ \int \left( \frac{z + \left| z\right| }{2}\right) < \infty \]\n\nand\n\n\[ \bar{\int }\left( z\right) = \bar{\int }\left( \frac{z + \left| z\right| }{2}\right) + \bar{\int }\left( \frac{z - \left| z\right| }{2}\right) .... | Proof. Suppose \( \bar{\int }\left( z\right) < P < \infty \) . Find \( m \in \mathcal{C} \) and \( \left( {x}_{n}\right) \subseteq \mathcal{C} \) so that \( {x}_{n} \geq m \) for all \( n \), and \( \mathop{\liminf }\limits_{n}{x}_{n} \geq z,\mathop{\lim }\limits_{n}f\left( {x}_{n}\right) < P \) . Notice that if \( {x}... | Yes |
Lemma 4.18. If \( z \in \mathcal{L} \), then \( \left| z\right| \in \mathcal{L} \), that is, \( \mathcal{L} \) is a vector lattice. | Proof. Since\n\n\[ \left| z\right| = \left( \frac{z + \left| z\right| }{2}\right) + \left( \frac{\left| z\right| - z}{2}\right) \]\n\nit’s enough (thanks to \( \mathcal{L} \) ’s linearity) to show that \( \frac{z + \left| z\right| }{2},\frac{z - \left| z\right| }{2} \) both belong to \( \mathcal{L} \) if \( z \in \math... | Yes |
Lemma 4.20 (Dominated Convergence Theorem). Suppose \( \left( {z}_{n}\right) \subseteq \mathcal{L}, M \in \) \( \mathcal{L} \) and \( \left| {z}_{n}\right| \leq M \) . Then\n\n\[ g = \mathop{\liminf }\limits_{n}{z}_{n}, h = \mathop{\limsup }\limits_{n}{z}_{n} \in \mathcal{L} \]\n\nwith\n\n\[ \bar{\int }\left( g\right) ... | Proof. For each \( i \) and for each \( j \geq i \), write\n\n\[ {g}_{ij} = \min \left\{ {{z}_{i},{z}_{i + 1},\ldots ,{z}_{j}}\right\} \]\n\nThen the sequence \( {\left( {g}_{ij}\right) }_{j = i}^{\infty } \) is decreasing, each member belongs to \( \mathcal{L} \), and so the sequence \( {\left( M - {g}_{ij}\right) }_{... | Yes |
Lemma 4.21. If \( z \in \mathcal{L}, z \geq 0 \) and \( \bar{\int }z = 0 \), then whenever the function \( x \) satisfies \( \left| x\right| \leq z \), we have that \( x \in \mathcal{L} \) and \( \bar{\int }x = 0 \) . | This is an immediate consequence of Lemma 4.15. | No |
Theorem 4.25. \( \mathcal{L} \) contains all of the indicator functions on Borel sets. | Suppose \( \left( {B}_{n}\right) \) is a sequence of pairwise disjoint Borel subsets of \( Q \) . Then \( {\chi }_{{B}_{n}},{\chi }_{\mathop{\bigcup }\limits_{n}{B}_{n}} \in \mathcal{L} \) and\n\n\[{\chi }_{\mathop{\bigcup }\limits_{n}{B}_{n}} = \mathop{\sum }\limits_{n}{\chi }_{{B}_{n}}\]\n\nan appeal to \( \left( {\m... | Yes |
Theorem 4.26 (Hausdorff’s Paradox). The unit sphere \( {S}^{2} \) of \( {\mathbb{R}}^{3} \) can be decomposed into the disjoint union\n\n\[ \n{S}^{2} = Q \cup R \cup S \cup T \n\]\n\nof four sets: \( Q \) is countable, \( R, S, T \) are pairwise congruent, and \( R \) is congruent to \( S \cup T \) . | Now suppose that \( \mu \) is a solution to the easy problem of measure in \( {\mathbb{R}}^{3} \) . Normalize \( \mu \) so \( \mu \left( {B}_{{\mathbb{R}}^{3}}\right) = 1 \) . With Hausdorff’s paradox in hand, write \( {B}_{0} = {B}_{{\mathbb{R}}^{3}} \smallsetminus \{ 0\} \) in the form\n\n\[ \n{B}_{0} = {Q}_{0} \cup ... | No |
Theorem 4.32. Let \( \Phi \) be a hyperfunction not belonging to the linear space \( \Omega \) of hyperfunctions, and suppose that there exist \( {F}_{1},{F}_{2} \in \Omega \) such that\n\n\[ \n{F}_{1} > \Phi > {F}_{2} \n\]\n\nLet \( A : \Omega \rightarrow \mathbb{R} \) be a positive linear functional. Then there exist... | The hypothesis of Theorem 4.32 tells us that\n\n\[ \n\alpha = \inf \{ A\left( F\right) : F \in \Omega, F > \Phi \} \n\]\n\nis real. Now if \( \Psi \in \Omega \left( \Phi \right) \) is written in the form\n\n\[ \n\Psi = F + {c\Phi } \n\]\n\nwhere \( c \in \mathbb{R}, F \in \Omega \), then\n\n\[ \n\bar{A}\left( \Psi \rig... | Yes |
Theorem 4.35. Suppose \( f \) is Riemann integrable. Then \( f \sim c \) where \( c = \) \( {\int }_{0}^{1}f\left( x\right) {dx} \) | Proof. Suppose \( f \) is Riemann integrable, and define \( {f}_{n} \) on \( C \) by\n\n\[ \n{f}_{n}\left( x\right) = \frac{1}{n}\mathop{\sum }\limits_{{k = 1}}^{n}f\left( {x + \frac{k}{n}}\right) = \frac{1}{n}\mathop{\sum }\limits_{{k = 1}}^{n}{f}_{\frac{k}{n}}\left( x\right) .\n\]\n\nThen \( \left( {f}_{n}\right) \) ... | Yes |
Proposition 4.37. There exists a bounded Lebesgue integrable function \( \rho \) with \( {\int }_{0}^{1}\rho \left( x\right) {dx} = 0 \) such that if \( \phi \) is Riemann integrable and \( \phi \succ \rho \), then \( {\int }_{0}^{1}\phi \left( x\right) {dx} > 1 \) . | If \( N \) is a set of Lebesgue measure zero with \( {N}^{c} \) a set of the first Baire category, then \( \rho = {\chi }_{N} \), the indicator function of \( N \), is bounded, Lebesgue integrable, and has \( {\int }_{0}^{1}\rho \left( x\right) {dx} = 0 \) . This \( \rho \) fits the bill. | Yes |
Theorem 5.1. Let \( G \) be a topological group, and let \( M \) be a nonempty compact subset of \( G \) . Then any continuous function \( f : G \rightarrow \mathbb{R} \) is left uniformly continuous on \( M \) ; i.e., given \( \epsilon > 0 \), there is an open set \( V \) containing the identity of \( G \) so that if ... | Proof. Let \( \epsilon > 0 \) be given. For each \( a \in M \), there is an open set \( {V}_{a} \) that contains the identity such that if \( x \in M \) and \( x \in a{V}_{a} \), then \( \left| {f\left( x\right) - f\left( a\right) }\right| \leq \frac{\epsilon }{2} \) . Since \( e \cdot e = e \), there is an open set \(... | Yes |
Theorem 5.4 (Arzelá and Ascoli). A nonempty subset \( \mathcal{K} \) of \( C\left( S\right) \) is relatively norm compact if and only if \( \mathcal{K} \) is uniformly bounded and equicontinuous. | Proof. Assume \( \mathcal{K} \) is uniformly bounded and equicontinuous. By Fact 5.3 the pointwise closure, \( \overline{{\mathcal{K}}^{p}} \) of \( \mathcal{K} \) is also equicontinuous. But \( \mathcal{K} \) is bounded in each coordinate by its uniform bound, and so \( \overline{{\mathcal{K}}^{p}} \) is also bounded ... | Yes |
Corollary 5.5. Let \( G \) be a topological group, and let \( M \) be a nonempty compact subset of \( G \) . Suppose \( \mathcal{K} \subseteq C\left( G\right) \) is equicontinuous on \( M \) . Then given \( \epsilon > 0 \) there is an open set \( V \) containing the identity of \( G \) so that if \( x, y \in M \) and \... | Proof. (It is useful to take a close look at the proof of Theorem 5.1 before looking at this proof.)\n\nLet \( \epsilon > 0 \) be given. For each \( a \in M \) there is an open set \( {V}_{a} \) that contains the identity \( e \) of \( G \) such that if \( x \in M \) and \( x \in a{V}_{a} \), then \( \left| {f\left( x\... | Yes |
Lemma 5.8. If \( f \in C\left( G\right) \) is not constant, then there is an \( F \in \mathcal{F}\left( G\right) \) such that\n\n\[ \n{\mathrm{{OscRAve}}}_{F}f < \mathrm{{Osc}}f.\n\] | Proof. After all, \( f \) ’s not being constant ensures that there is an \( \alpha \) such that \( \min f < \alpha < \max f \) . Set\n\n\[ \nU = \left\lbrack {f < \alpha }\right\rbrack = \{ x \in G : f\left( x\right) < \alpha \} .\n\]\n\nSince \( \min f < \alpha ,\;U \) is a nonempty open set in \( G \) and \( G = \mat... | Yes |
Lemma 5.9. Let \( f \in C\left( G\right) \), and define \( \mathcal{K} = \left\{ {{\operatorname{RAve}}_{F}f : F \in \mathcal{F}\left( G\right) }\right\} \) . Then \( \mathcal{K} \) is uniformly bounded, equicontinuous family in \( C\left( G\right) \) . | Proof. The key to this precious fact is that \( f \) is of course uniformly continuous. So given an \( \epsilon > 0 \), there is an open set \( V \) in \( G \) containing \( G \) ’s identity such that if \( x{y}^{-1} \in V \), then \( \left| {f\left( x\right) - f\left( y\right) }\right| \leq \epsilon \) . Notice that i... | Yes |
Lemma 5.10. Let \( f \in C\left( G\right) \) and \( \mathcal{K} = \left\{ {{\operatorname{RAve}}_{F}f : F \in \mathcal{F}\left( G\right) }\right\} \) . Then\n\n\[ \mathop{\inf }\limits_{{g \in \mathcal{K}}}\operatorname{Osc}g = 0 \] | Proof. Let\n\n\[ s = \mathop{\inf }\limits_{{g \in \mathcal{K}}}\operatorname{Osc}g = \inf \left\{ {{\operatorname{OscRAve}}_{F}f : F \in \mathcal{F}\left( G\right) }\right\} .\n\nTherefore there exists \( \left( {F}_{n}\right) \) in \( \mathcal{F}\left( G\right) \) such that Osc \( \left( {\text{RAve}}_{{F}_{n}}\right... | Yes |
Theorem 5.11. Every \( f \in C\\left( G\\right) \) has a right mean. | Proof. By the techniques used in Lemma 5.10, there is a constant function \( h \) (say \( h\\left( x\\right) \\equiv p \) ) and a sequence \( \\left( {F}_{n}\\right) \\subseteq \\mathcal{F}\\left( G\\right) \) such that\n\n\[ \n\\mathop{\\lim }\\limits_{n}{\\begin{Vmatrix}{\\operatorname{RAve}}_{{F}_{n}}f - h\\end{Vmat... | Yes |
Theorem 5.12. Let \( f \in C\left( G\right) \) . Let \( p \) be a right mean of \( f \), and let \( q \) be a left mean of \( f \) . Then \( p = q \) . | Proof. Let \( \epsilon > 0 \) . Find \( A, B \in \mathcal{F}\left( G\right) \) so that\n\n\[ \left| \right| {\mathrm{{RAve}}}_{A}f - p{\left| \right| }_{\infty } \leq \frac{\epsilon }{2},\;\left| \right| {\mathrm{{LAve}}}_{B}f - q{\left| \right| }_{\infty } \leq \frac{\epsilon }{2}. \]\n\nNow\n\n\[ {\operatorname{RAve}... | Yes |
Theorem 5.15 (L. Kantorovitch). Let \( {x}^{ * } \) be a positive linear functional on \( C\left( K\right) \) . Then there is a positive linear functional \( {y}^{ * } \) on \( {l}^{\infty }\left( K\right) \) which is an extension of \( {x}^{ * } \) without increasing its norm. | The proof relies on the fact that \( C{\left( K\right) }^{ * } \) \ | No |
Theorem 5.17 (Alexandroff). Let \( \mu \) be a regular finitely additive nonnegative measure defined on the Borel field \( \mathcal{A} \) (the field generated by the open sets) of the compact Hausdorff space \( K \) . Then \( \mu \) is countably additive on \( \mathcal{A} \) . | Proof. Let \( \epsilon > 0 \) . Suppose \( \left( {E}_{n}\right) \) is a sequence of pairwise disjoint members of \( \mathcal{A} \) with \( E = \mathop{\bigcup }\limits_{n}{E}_{n} \in \mathcal{A} \) . There is an \( F \in \mathcal{A}, F \) closed, \( F \subseteq E \), with \( \mu \left( {E \smallsetminus F}\right) < \e... | Yes |
Theorem 5.18. With bigamy an anathema, for each boy in \( B \) to be able to marry a girl with whom he is acquainted it is both necessary and sufficient that regardless of \( C \subseteq B \)\n\n\[ \left| {\mathop{\bigcup }\limits_{{b \in C}}G\left( b\right) }\right| \geq \left| C\right| \] holds. | Proof. Since it is plain that (25) is necessary, we'll concentrate on proving sufficiency of (25). So we suppose (25) to be in effect and prove the possibility of a wise matchmaker by an induction on the number of boys in \( B \) . Though not universally associated with marriage, we nevertheless introduce the notion of... | No |
Lemma 5.20. If \( C \) and \( {C}^{\prime } \) are perfect, then so is \( C \cup {C}^{\prime } \) . | Proof. By (25) we know that\n\n\[ \left| {\mathop{\bigcup }\limits_{{b \in C \cup {C}^{\prime }}}G\left( b\right) }\right| \geq \left| {C \cup {C}^{\prime }}\right| \]\n\nFurther\n\n\[ \left| {C \cup {C}^{\prime }}\right| = \left| C\right| + \left| {C}^{\prime }\right| - \left| {C \cap {C}^{\prime }}\right| \]\n\nand, ... | Yes |
Theorem 5.21 (Markov-Kakutani Fixed Point Theorem). Suppose \( K \) is a compact convex subset of the locally convex (Hausdorff) linear topological space \( E \), and suppose that \( T \) is a continuous linear operator \( T : E \rightarrow E \) such that \( T\left( K\right) \subseteq K \) . Then there is an element \(... | Proof. Fix \( {k}_{0} \in K \), and set\n\n\[ {k}_{n} = \frac{1}{n + 1}\left( {{k}_{0} + T\left( {k}_{0}\right) + \cdots + {T}^{n}\left( {k}_{0}\right) }\right) . \]\n\nEach \( {k}_{n} \in K \) . Because \( K \) is compact the sequence \( \left( {k}_{n}\right) \) has a limit point \( k \) . Let \( \phi \in {E}^{ * } \)... | Yes |
Corollary 5.22. Suppose \( \mathcal{A} \) is a commuting family of continuous linear operators on the locally convex (Hausdorff) linear topological space, \( K \) is a compact convex subset of \( E \), and \( T\left( K\right) \subseteq K \) for each \( T \in \mathcal{A} \). Then there is \( {ak} \in K \) such that \[ T... | Proof. First we notice that if \( {T}_{1},\ldots ,{T}_{n} \in \mathcal{A} \), then there is a \( k \in K \) so that \[ {T}_{1}\left( k\right) = k,{T}_{2}\left( k\right) = k,\ldots ,{T}_{n}\left( k\right) = k. \] This follows by induction on \( n \). The Markov-Kakutani theorem shows it so for \( n = 1 \). Suppose we kn... | Yes |
Corollary 5.25. Let \( G \) be a compact topological group. Then there exists a regular Borel probability \( \mu \) on \( G \) such that\n\n\[ \mu \left( {{g}^{-1}A}\right) = \mu \left( A\right) \]\n\nfor each \( g \in G \) and each Borel set \( A \subseteq G \) . | Proof. The proof proceeds along the same path as that outlined in the case when \( G \) is abelian. This time we see that the collection \( \left\{ {{T}_{g} : g \in G}\right\} \) is a group of isometries of \( C\left( G\right) \) onto \( C\left( G\right) \) and, as before,\n\n\[ {T}_{g}^{ * }\left( \mu \right) = {\mu }... | Yes |
Theorem 6.1 (Weil). Let the compact group \( G \) act transitively on the compact Hausdorff space \( K \) . Then there is a closed subgroup \( H \) of \( G \) such that \( K \) and \( G/H \) are isomorphic under \( G \)’s action. | Proof. Fix \( {k}_{0} \in K \) . Look at\n\n\[ H = \left\{ {g \in G : g\left( {k}_{0}\right) = {k}_{0}}\right\} \]\n\n\( H \) is called an isotopy subgroup. It is plain that \( H \) is a closed subgroup of \( G \) . A natural candidate for the isomorphism of \( G/H \) and \( K \) is at hand: \( \phi : G/H \rightarrow K... | Yes |
Theorem 6.2 (Weil). Suppose the compact group \( G \) acts transitively on the compact Hausdorff space \( K \) . Then there is a unique \( G \) -invariant regular Borel probability measure on \( K \) . | Proof. We identify \( K \) with \( G/H \) where \( H \) is an isotopy subgroup of \( G \) as in our previous theorem. Let\n\n\[ \n{q}_{H} : G \rightarrow G/H \n\] \n\nbe the natural quotient map. Suppose \( \mu \) is the normalized Haar measure on \( G \) and define \( {\mu }_{G/H} \) on \( G/H \) by \n\n\[ \n{\mu }_{G... | Yes |
Lemma 6.8 (Schur’s Lemma). Let \( E \) and \( F \) be finite-dimensional complex linear spaces, and suppose \( \mathcal{R} \) and \( \mathcal{S} \) are irreducible families of linear transformations on \( E \) and \( F \), respectively. Assume that \( T : E \rightarrow F \) is a linear transformation such that given an... | Proof. Step One. Suppose \( \dim E = m,\dim F = n \), and \( \left\{ {{e}_{1},\ldots ,{e}_{m}}\right\} \) , \( \left\{ {{f}_{1},\ldots ,{f}_{n}}\right\} \) are a bases for \( E \) and \( F \), respectively. Let\n\n\[ \n{b}_{j} = T{e}_{j},\text{ and }L = \operatorname{span}\left\{ {{b}_{1},\ldots ,{b}_{m}}\right\} = T\l... | Yes |
Corollary 6.9. Let \( E \) be a finite-dimensional complex linear space, and let \( \mathcal{R} \) be an irreducible family of linear transformations on \( E \) . Suppose \( A \) is a linear transformation on \( E \) such that\n\n\[ \n{AB} = {BA} \n\]\n\nfor each \( B \in \mathcal{R} \) . Then \( A = \alpha {\operatorn... | Proof. If \( A \neq 0 \) and \( \alpha \) is a nonzero eigenvalue of \( A \), then \( A - \alpha {\operatorname{id}}_{E} \) commutes with each \( B \in \mathcal{R} \) . Schur’s Lemma tells us either \( A - \alpha {\operatorname{id}}_{E} = 0 \) or \( A - \alpha {\operatorname{id}}_{E} \) is invertible. The latter is not... | Yes |
Theorem 6.11. Let \( G \) be a compact topological group with normalized Haar measure \( \mu \) . Suppose \( x \rightarrow {U}_{x} = \left( {{u}_{{i}_{j}}\left( x\right) }\right) \) and \( x \rightarrow {V}_{x} = \left( {{v}_{kl}\left( x\right) }\right) \) are nonequivalent irreducible unitary representations of \( G \... | \[ {\int }_{G}{u}_{{i}_{j}}\left( x\right) \overline{{v}_{kl}\left( x\right) }{d\mu }\left( x\right) = 0 \] Proof. Let \( T : F \rightarrow E \) be a linear transformation. For any \( x \in G \), let \[ {A}_{x} = {U}_{x}T{V}_{x}^{-1} = {U}_{x}T{V}_{{x}^{-1}} = {U}_{x}T{V}_{x}^{ * }. \] Each \( {A}_{x} \) is a linear tr... | Yes |
Theorem 6.12. The net \( {\left( {\kappa }_{U} * f\right) }_{U} \) converges to \( f \) in \( C\left( G\right) \) for each \( f \) . | Proof. Let \( f \in C\left( G\right) \) and \( \epsilon > 0 \) be given.\n\nBy \( f \) ’s uniform continuity, there is an open set \( W \) containing \( e \) such that regardless of \( s \in G \) if \( t \in W \), then\n\n\[ \left| {f\left( {{t}^{-1}s}\right) - f\left( s\right) }\right| \leq \epsilon \]\n\nLet \( U \) ... | No |
Theorem 6.13 (Pietsch). Let \( u : X \rightarrow Y \) be a bounded linear operator between Banach spaces \( X \) and \( Y \) . Let \( K \subseteq {B}_{{X}^{ * }} \) be a weak*-compact, norming set. Then \( u \) is an absolutely p-summing operator if and only if there exists a \( \textit{regular Borel probability measur... | Pietsch's original proof was somewhat complicated, but in the early seventies a shorter proof was offered by B. Maurey [81]. We present an outline of Maurey's proof of the existence of the \ | No |
Corollary 7.8. If \( {f}_{1},{f}_{2} \in {\mathcal{L}}^{ + } \), then \( \bar{I}\left( {{f}_{1} + {f}_{2}}\right) = \bar{I}\left( {f}_{1}\right) + \bar{I}\left( {f}_{2}\right) \) . | This is an easy consequence of \( \phi \) ’s additivity on \( {\mathcal{K}}^{ + }\left( \mathcal{S}\right) \), and the fact that\n\n\[ \n{f}_{1} + {f}_{2} = \sup \left\{ {{g}_{1} + {g}_{2} : {g}_{1},{g}_{2} \in {\mathcal{K}}^{ + },{g}_{1} \leq {f}_{1},{g}_{2} \leq {f}_{2}}\right\} \n\] | Yes |
Corollary 7.9. For any nonempty \( \mathcal{D} \subseteq {\mathcal{L}}^{ + } \) , | \[ \bar{I}\left( {\mathop{\sum }\limits_{{f \in \mathcal{D}}}f}\right) = \mathop{\sum }\limits_{{f \in \mathcal{D}}}\bar{I}\left( f\right) \] | Yes |
Theorem 7.12. If \( \left( {f}_{n}\right) \) is an ascending sequence of functions with \( {f}_{n} \) : \( S \rightarrow \left\lbrack {0,\infty }\right\rbrack \), then \[ {\int }_{\begin{matrix} {\text{ sup }{f}_{n} = \sup } \\ n \end{matrix}}{f}_{n} \] | Proof. It's plain from Fact 7.11(iii) that \[ \int \mathop{\sup }\limits_{n}{f}_{n} \geq \mathop{\sup }\limits_{n}\int {f}_{n} \] so we will establish the reverse inequality, namely, \[ \mathop{\sup }\limits_{n}\int {f}_{n} \geq {\int }_{n}\mathop{\sup }\limits_{n}{f}_{n} \] If \( \mathop{\sup }\limits_{n}\bar{\int }{f... | Yes |
Theorem 7.15. (i) If \( \mathcal{U} \) is a family of pairwise disjoint open sets, then \[ \mu \left( {\mathop{\bigcup }\limits_{\mathcal{U}}U}\right) = \mathop{\sum }\limits_{\mathcal{U}}\mu \left( U\right) \] | Proof. To see item (i): if \( U \) is open, then \( {\chi }_{U} \in {\mathcal{L}}^{ + } \) . Indeed, if \( {s}_{0} \in U \) , then \( 1 = {\chi }_{U}\left( {s}_{0}\right) > h \) also holds for any \( s \in U \) ; i.e., \( {\chi }_{U}\left( s\right) > h \) . If \( {s}_{0} \notin U \) , then \( {\chi }_{U}\left( {s}_{0}\... | No |
Theorem 7.17. All Borel subsets of \( S \) are \( \mu \) -measurable. | Proof. Suppose \( V \) is an open subset of \( \mathcal{S} \). Let \( U \) be an open subset in \( \mathcal{S} \) with \( \mu \left( U\right) < \infty \), and let \( \epsilon > 0 \) be given. Use Theorem 7.15 to find an open subset \( O \) of \( \mathcal{S} \) so that\n\n\[ U \cap {V}^{c} \subseteq O \]\n\nand\n\n(31)\... | Yes |
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