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Proposition 13.3 Assume that \( \{ X,\mathcal{A},\mu \} \) and \( \{ Y,\mathcal{B},\nu \} \) are complete measure spaces, and Let \( E \in \left( {\mathcal{A} \times \mathcal{B}}\right) \) be of \( \left( {\mu \times \nu }\right) \) -measure zero. Then\n\n\[ \n{E}_{x}\text{are}\nu \text{-measurable and}\nu \left( {E}_{... | Proof If \( E \in {\mathcal{R}}_{\sigma \delta } \) the conclusion follows from the previous proposition. If \( E \) is \( \left( {\mu \times \nu }\right) \) -measurable, there exist a set \( {E}_{\sigma \delta } \in {\mathcal{R}}_{\sigma \delta } \) such that \( E \subset {E}_{\sigma \delta } \) and \( (\mu \times \) ... | Yes |
Proposition 13.4 Assume that \( \{ X,\mathcal{A},\mu \} \) and \( \{ Y,\mathcal{B},\nu \} \) are complete measure spaces and let \( E \in \left( {\mathcal{A} \times \mathcal{B}}\right) \) be of finite \( \left( {\mu \times \nu }\right) \) -measure. Then\n\n\( {E}_{x} \) are \( \nu \) -measurable for \( \mu \) -a.e. \( ... | Proof There exists \( {E}_{\sigma \delta } \in {\mathcal{R}}_{\sigma \delta } \), such that \( E \subset {E}_{\sigma \delta } \) and \( {E}_{\sigma \delta } - E = \mathcal{E} \) has \( \left( {\mu \times \nu }\right) \) - measure zero. Therefore, by Proposition 13.3 the sets \( {E}_{x} = {E}_{{\sigma \delta }, x} - {\m... | Yes |
Theorem 14.1 (Fubini [52]) Let \( \{ X,\mathcal{A},\mu \} \) and \( \{ Y,\mathcal{B},\nu \} \) be two complete measure spaces and let\n\n\[ \nX \times Y \ni \left( {x, y}\right) \rightarrow f\left( {x, y}\right) \;\text{ be integrable in }X \times Y.\n\]\n\nThen\n\n\[ \nX \ni x \rightarrow f\left( {x, y}\right) \text{i... | Proof By the decomposition (3.4) we may assume that \( f \geq 0 \) . By Proposition 13.4, the statement holds true if \( f \) is the characteristic function of a measurable set \( E \) of finite measure. If \( f \) is nonnegative and integrable there exists a sequence \( \left\{ {f}_{n}\right\} \) of nonnegative integr... | Yes |
Theorem 14.2 (Tonelli [160]) Let \( \{ X,\mathcal{A},\mu \} \) and \( \{ Y,\mathcal{B},\nu \} \) be complete and \( \sigma \) -finite, and let \( f : \left( {X \times Y}\right) \rightarrow {\mathbb{R}}^{ * } \) be measurable and nonnegative. Then the measurability statements in (14.1) and the double integral formula (1... | Proof The integrability requirement in the Fubini theorem was used to insure the existence of a sequence \( \left\{ {f}_{n}\right\} \) of integrable functions each vanishing outside a set of finite measure and converging to \( f \) . The positivity of \( f \) and the \( \sigma \) -finiteness in the Tonelli theorem prov... | Yes |
Proposition 15.1 Let \( \{ X,\mathcal{A},\mu \} \) be complete and \( \sigma \) -finite and let \( f : E \rightarrow {\mathbb{R}}^{ * } \) be measurable and nonnegative. Let also \( \nu \) be a complete and \( \sigma \) -finite measure on \( {\mathbb{R}}^{ + } \) such that \( \nu \left( {\lbrack 0, t}\right) ) = \nu \l... | Proof The function \( f : E \rightarrow {\mathbb{R}}^{ * } \), when regarded as a function from \( E \times {\mathbb{R}}^{ + } \) into \( {\mathbb{R}}^{ * } \), is measurable in the product measure \( \left( {\mu \times \nu }\right) \) . Likewise, the function \( g\left( t\right) = t \) from \( {\mathbb{R}}^{ + } \) in... | Yes |
Corollary 15.1 Let \( E \) be an open set in \( {\mathbb{R}}^{N} \) and let \( f \) be continuous in \( E \) . Then for all \( x \in E \) , \[ f\left( x\right) = {\int }_{0}^{\infty }{\chi }_{\left\lbrack f\left( x\right) > 0\right\rbrack }{dt}. \] | Proof Apply (15.2) with \( \mu = {\delta }_{x} \) . | No |
Corollary 15.2 Let \( E \) be an open set in \( {\mathbb{R}}^{N} \) and let \( f \) and \( h \) be nonnegative Lebesgue measurable functions defined in \( E \) . Then for all \( p > 0 \)\n\n\[{\int }_{E}{f}^{p}{hdx} = {\int }_{0}^{\infty }p{t}^{p - 1}\left( {{\int }_{\left\lbrack f > t\right\rbrack }{hdx}}\right) {dt}\... | Proof Apply (15.2) with \( {d\mu } = {hdx} \). | No |
Lemma 15.1 Let \( f : {\mathbb{R}}^{N} \rightarrow {\mathbb{R}}^{ * } \) be measurable. Then the function\n\n\[ \n{\mathbb{R}}^{2N} \ni \left( {x, y}\right) \rightarrow f\left( {x - y}\right) \n\]\n\nis measurable with respect to the product measure of \( {\mathbb{R}}^{2N} \) . | Proof Consider the change of variables\n\n\[ \n{\mathbb{R}}^{2N} \ni \left( \begin{array}{l} x \\ y \end{array}\right) \rightarrow T\left( \begin{array}{l} x \\ y \end{array}\right) = \left( \begin{array}{l} x - y \\ x + y \end{array}\right) = \left( \begin{array}{l} \xi \\ \eta \end{array}\right) \in {\mathbb{R}}^{2N}... | Yes |
Proposition 15.2 Let \( f \) and \( g \) be nonnegative and integrable in \( {\mathbb{R}}^{N} \). Then \( \left( {f * g}\right) \left( x\right) \) is finite for a.e. \( x \in {\mathbb{R}}^{N} \), the function \( \left( {f * g}\right) \) is integrable in \( {\mathbb{R}}^{N} \) and\n\n\[{\int }_{{\mathbb{R}}^{N}}\left( {... | Proof The function \( \left( {x, y}\right) \rightarrow g\left( y\right) f\left( {x - y}\right) \) is nonnegative and measurable with respect to the product measure. Therefore, by the Tonelli theorem\n\n\[{\iint }_{{\mathbb{R}}^{2N}}g\left( y\right) f\left( {x - y}\right) {dxdy} = {\int }_{{\mathbb{R}}^{N}}\left( {{\int... | Yes |
Lemma 15.2 Let \( E \) be a nonempty set in \( {\mathbb{R}}^{N} \). Then the distance function \( x \rightarrow \delta \left( x\right) \) is Lipschitz continuous with Lipschitz constant one. | Proof Fix \( x \) and \( y \) in \( {\mathbb{R}}^{N} \) and assume that \( {\delta }_{E}\left( x\right) \geq {\delta }_{E}\left( y\right) \). By definition of \( {\delta }_{E}\left( y\right) \), having fixed \( \varepsilon > 0 \) there exists \( {z}^{\prime } \in E \) such that \( {\delta }_{E}\left( y\right) \geq \lef... | Yes |
Proposition 15.3 The Marcinkiewicz integral \( {M}_{E,\lambda }\left( x\right) \) is finite for a.e. \( x \in E \) . Moreover the function \( x \rightarrow {M}_{E,\lambda }\left( x\right) \) is integrable in \( E \) and\n\n\[{\int }_{E}{M}_{E,\lambda }{dx} \leq \frac{{\omega }_{N}}{\lambda }\mu \left( {Q - E}\right)\] | Proof Since \( {\delta }_{E}\left( y\right) = 0 \) for all \( y \in E \), by the Tonelli theorem\n\n\[{\int }_{E}{M}_{E,\lambda }\left( x\right) {dx} = {\int }_{Q}{\delta }_{E}^{\lambda }\left( y\right) \left( {{\int }_{E}\frac{dx}{{\left| x - y\right| }^{N + \lambda }}}\right) {dy}\]\n\n(15.7)\n\n\[= {\int }_{Q - E}{\... | Yes |
Lemma 16.1 Let \( \{ X,\mathcal{A},\mu \} \) be a measure space for a signed measure \( \mu \) . Let \( E \subset X \) be measurable and such that \( \left| {\mu \left( E\right) }\right| < \infty \) . Then every measurable set \( A \subset E \) satisfies \( \left| {\mu \left( A\right) }\right| < \infty \) . | Proof Assume for example that \( \mu \) does not take the value \( + \infty \) . Let \( A \subset E \) . If \( \mu \left( A\right) > 0 \) then \( \mu \left( A\right) < \infty \) . If \( \mu \left( A\right) < 0 \), taking the measure of the disjoint union \( E = \left( {E - A}\right) \cup A \) gives \( \mu \left( E\righ... | Yes |
Proposition 16.1 Let \( \{ X,\mathcal{A},\mu \} \) be a measure space for a signed measure \( \mu \) . Every measurable set \( E \) of positive, finite measure contains a positive subset \( A \) of positive measure. | Proof If \( E \) is positive we take \( A = E \) . Otherwise \( E \) contains a measurable set of negative measure. Let \( {n}_{1} \) be the smallest positive integer for which there exists a measurable set \( {B}_{1} \subset E \) such that\n\n\[ \mu \left( {B}_{1}\right) \leq - \frac{1}{{n}_{1}} \]\n\nIf \( {A}_{1} = ... | Yes |
Theorem 16.1 (Hahn Decomposition [58], Vol. 1) Let \( \{ X,\mathcal{A},\mu \} \) be a measure space for a signed measure \( \mu \) . Then \( X \) can be decomposed into a positive set \( {X}^{ + } \) and a negative set \( {X}^{ - } \) . | Proof Assume for example that \( \mu \) does not take the value \( + \infty \) and set\n\n\[ M = \{ \sup \mu \left( A\right) \text{ where }A \in \mathcal{A}\text{ is positive }\} .\n\]\n\nLet \( \left\{ {A}_{n}\right\} \) be a sequence of positive sets such that \( \mu \left( {A}_{n}\right) \) increases to \( M \) and ... | Yes |
Proposition 17.1 (von Neumann [114]) For a countable index \( t \), let \( \left\{ {E}_{t}\right\} \) be a collection of measurable sets such that \( {E}_{s} \subset {E}_{t} \) for \( s < t \) . There exists a measurable function \( f : X \rightarrow {\mathbb{R}}^{ * } \) such that\n\n\[ f \leq t\text{ in }{E}_{t},\;\t... | Proof Define\n\n\[ X \ni x \rightarrow f\left( x\right) = \left\{ \begin{array}{ll} \inf \left\{ {t \mid x \in {E}_{t}}\right\} & \text{ if }x \in \bigcup {E}_{t} \\ + \infty & \text{ otherwise. } \end{array}\right. \]\n\nIf \( x \in {E}_{t} \), then \( f\left( x\right) \leq t \) . If \( x \notin {E}_{t} \), then \( x ... | Yes |
For a countable index \( t \), let \( \left\{ {E}_{t}\right\} \) be a collection of measurable sets such that\n\n\[ \mu \left( {{E}_{s} - {E}_{t}}\right) = 0\;\text{ whenever }s < t. \]\n\nThere exists a measurable function \( f : X \rightarrow {\mathbb{R}}^{ * } \)\n\n\[ f \leq t\text{ a.e. in }{E}_{t},\;\text{ and }\... | Proof Set\n\n\[ \mathcal{E} = \mathop{\bigcup }\limits_{{s < t}}\left( {{E}_{s} - {E}_{t}}\right) \;\text{ and }\;{E}_{t}^{\prime } = {E}_{t} \cup \mathcal{E}. \]\n\nThe collection \( \left\{ {E}_{t}^{\prime }\right\} \) is monotone and, by Proposition 17.1, there exists a measurable function \( f : X \rightarrow {\mat... | Yes |
Theorem 18.2 (Lebesgue) Let \( \{ X,\mathcal{A},\mu \} \) be a \( \sigma \) -finite measure space for a signed measure \( \mu \) and let \( \nu \) be a \( \sigma \) -finite signed measure defined on \( \mathcal{A} \) . There exists a unique pair \( \left( {{\nu }_{o},{\nu }_{1}}\right) \) of \( \sigma \) -finite, signe... | Proof If \( \nu = {\nu }_{o}^{\prime } + {\nu }_{1}^{\prime } \) is another such decomposition, then\n\n\[ {\nu }_{o} - {\nu }_{o}^{\prime } = {\nu }_{1}^{\prime } - {\nu }_{1} \]\n\nThis implies that \( {\nu }_{o} - {\nu }_{o}^{\prime } \) is both singular and absolutely continuous with respect to \( \mu \) and theref... | Yes |
Proposition 1.3 Let \( f \) be of bounded variation in \( \left\lbrack {a, b}\right\rbrack \) . Then\n\n\[ \n{V}_{f}\left\lbrack {a, b}\right\rbrack = {V}_{f}^{ + }\left\lbrack {a, b}\right\rbrack + {V}_{f}^{ - }\left\lbrack {a, b}\right\rbrack \n\]\n\n\[ \nf\left( b\right) - f\left( a\right) = {V}_{f}^{ + }\left\lbrac... | Since \( x \rightarrow {V}_{f}^{ \pm }\left\lbrack {a, x}\right\rbrack \) are both non-decreasing, a function \( f \) of bounded variation can be written as the difference of two non-decreasing functions. We have already observed that the difference of two monotone functions in \( \left\lbrack {a, b}\right\rbrack \) is... | Yes |
Proposition 1.4 A function \( f \) of bounded variation in \( \left\lbrack {a, b}\right\rbrack \) has at most countably many jump discontinuities in \( \left\lbrack {a, b}\right\rbrack \) . | Proof May assume that \( f \) is monotone increasing. Then, for every \( c \in \left( {a, b}\right) \), the limits\n\n\[ \n\mathop{\lim }\limits_{{x \rightarrow {c}^{ + }}}f\left( x\right) = f\left( {c}^{ + }\right) ,\;\mathop{\lim }\limits_{{x \rightarrow {c}^{ - }}}f\left( x\right) = f\left( {c}^{ - }\right) \n\]\n\n... | Yes |
Proposition 2.1 (Banach-Sierpiński [9,147]) Let \( f \) be real-valued and nondecreasing in \( \left\lbrack {a, b}\right\rbrack \) . Then the functions \( {D}_{ \pm }f \) and \( {D}^{ \pm }f \) are measurable. | Proof We prove that \( {D}^{ + }f \) is measurable, the arguments for the remaining ones being similar. For \( n \in \mathbb{N} \) set\n\n\[ \n{u}_{n}\left( x\right) = \mathop{\sup }\limits_{{0 < h < \frac{1}{n}}}\frac{f\left( {x + h}\right) - f\left( x\right) }{h}.\n\]\n\nSince \( {D}^{ + }f = \lim {u}_{n} \), it suff... | Yes |
Proposition 2.2 Let \( f \) be a real-valued, non-decreasing function in \( \left\lbrack {a, b}\right\rbrack \) . Then\n\n\[ f\left( b\right) - f\left( a\right) \geq {t\mu }\left( \left\lbrack {{D}^{\prime \prime }f > t}\right\rbrack \right) \;\text{ for all }t \in \mathbb{R}. \] | Proof The assertion is trivial if \( \mu \left( \left\lbrack {{D}^{\prime \prime }f > t}\right\rbrack \right) = 0 \) or if \( t \leq 0 \) . Assuming that\n\n\[ \mu \left( \left\lbrack {{D}^{\prime \prime }f > t}\right\rbrack \right) > 0\;\text{ and }\;t > 0 \]\n\nlet \( \mathcal{F} \) denote the family of all closed in... | Yes |
Corollary 2.1 Let \( f \) be a real-valued, non-decreasing function defined in \( \left\lbrack {a, b}\right\rbrack \) . Then \( {D}^{\prime \prime }f \) and \( {D}^{\prime }f \) are a.e. finite in \( \left\lbrack {a, b}\right\rbrack \) . | Proof For all \( t > 0 \)\n\n\[ \mu \left( \left\lbrack {{D}^{\prime }f = \infty }\right\rbrack \right) \leq \mu \left( \left\lbrack {{D}^{\prime \prime }f = \infty }\right\rbrack \right) \leq \mu \left( \left\lbrack {{D}^{\prime \prime }f > t}\right\rbrack \right) \leq \frac{f\left( b\right) - f\left( a\right) }{t}. \... | Yes |
Theorem 4.1 (Fubini [53]) \( {}^{2} \) Let \( \left\{ {f}_{n}\right\} \) be a sequence of real-valued, non-decreasing functions in \( \left\lbrack {a, b}\right\rbrack \) and assume that the series \( \sum {f}_{n} \) is convergent in \( \left\lbrack {a, b}\right\rbrack \) to a real-valued function \( f \) defined in \( ... | Proof By possibly replacing \( {f}_{n} \) with \( {f}_{n} - {f}_{n}\left( a\right) \), we may assume that \( {f}_{n}\left( a\right) = 0 \) and \( {f}_{n} \geq 0 \) . For \( n \in \mathbb{N} \) write\n\n\[ \nf = \mathop{\sum }\limits_{{i = 1}}^{n}{f}_{i} + {R}_{n}\;\text{ where }\;{R}_{n} = \mathop{\sum }\limits_{{j = n... | Yes |
Proposition 5.1 Let \( f \) be absolutely continuous in \( \left\lbrack {a, b}\right\rbrack \) . Then \( f \) is of bounded variation in \( \left\lbrack {a, b}\right\rbrack \) . | Proof Having fixed \( \varepsilon > 0 \) and the corresponding \( \delta \), partition \( \left\lbrack {a, b}\right\rbrack \) by points \( a = \) \( {x}_{o} < {x}_{1} < \cdots < {x}_{n} = b \) such that\n\n\[ \frac{1}{2}\delta < {x}_{i} - {x}_{i - 1} < \delta \;i = 1,\ldots, n.\]\n\nThe number \( n \) of intervals of t... | Yes |
Proposition 5.2 Let \( f \) be absolutely continuous in \( \left\lbrack {a, b}\right\rbrack \) . If \( {f}^{\prime } \geq 0 \) a.e. in \( \left\lbrack {a, b}\right\rbrack \) then \( f \) is non-decreasing in \( \left\lbrack {a, b}\right\rbrack \) . | Proof Fix \( \left\lbrack {\alpha ,\beta }\right\rbrack \subset \left\lbrack {a, b}\right\rbrack \) and set \( E = \left\lbrack {{f}^{\prime } \geq 0}\right\rbrack \cap \left\lbrack {\alpha ,\beta }\right\rbrack \) . By the assumption \( \mu \left( E\right) = \) \( \beta - \alpha \) . Having fixed \( \varepsilon > 0 \)... | Yes |
Proposition 6.1 (Lebesgue [91]) Let \( E \subset \left\lbrack {a, b}\right\rbrack \) be Lebesgue measurable. Then\n\n\[ \n{d}_{E}^{\prime } = 1\text{ a.e. in }E\;\text{ and }\;{d}_{E}^{\prime } = 0\text{ a.e. in }\left\lbrack {a, b}\right\rbrack - E.\n\] | Proof It suffices to prove the first of these. If \( E \) is open then \( {d}_{E}^{\prime } = 1 \) in \( E \) . Assume now that \( E \) is of the type of a \( {\mathcal{G}}_{\delta } \), i.e., there exists a countable collection of open sets \( \left\{ {E}_{n}\right\} \) such that \( {E}_{n + 1} \subset {E}_{n} \) and ... | Yes |
Proposition 7.1 Let \( f \) be Lebesgue integrable in \( \left\lbrack {a, b}\right\rbrack \) . Then \( {F}^{\prime } = f \), a.e. in \( \left\lbrack {a, b}\right\rbrack \) . | Proof Assume first that \( f \) is simple. Then if \( \left\{ {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right\} \) are the distinct values taken by \( f \), there exist disjoint, measurable sets \( {E}_{i} \subset \left\lbrack {a, b}\right\rbrack, i = 1,\ldots, n \) such that\n\n\[ f = \mathop{\sum }\limits_{{i = 1}}^{n}... | Yes |
Proposition 7.2 (Lebesgue [91]) Let \( f \) be absolutely continuous in \( \left\lbrack {a, b}\right\rbrack \) . Then \( {f}^{\prime } \) is integrable and\n\n\[ f\left( x\right) = f\left( a\right) + {\int }_{a}^{x}{f}^{\prime }\left( t\right) {dt}. \] | Proof Assume first that \( f \) is non-decreasing so that \( {f}^{\prime } \geq 0 \) a.e. in \( \left\lbrack {a, b}\right\rbrack \) . Defining \( f\left( x\right) = f\left( b\right) \) for \( x \geq b \), the limit\n\n\[ {f}^{\prime }\left( x\right) = \lim \frac{f\left( {x + \frac{1}{n}}\right) - f\left( x\right) }{\fr... | Yes |
Proposition 8.1 Let \( \mu \) and \( \nu \) be two Radon measures in \( {\mathbb{R}}^{N} \) and let \( {\mu }_{e} \) and \( {\nu }_{e} \) be their associated outer measures. For every \( t > 0 \) and every set\n\n\[ E \subset \left\lbrack {{D}_{\mu }^{ + }\nu \geq t}\right\rbrack \;\text{ there holds }\;{\mu }_{e}\left... | Proof In proving (8.2) assume first that the set \( E \) is bounded. Fix \( t > 0 \) and \( \varepsilon \in \left( {0, t}\right) \) . By the definition of \( {D}_{\mu }^{ + }\nu \left( x\right) \), for every \( x \in E \), there exists a ball \( {B}_{\rho }\left( x\right) \) centered at \( x \) and of arbitrarily small... | Yes |
Proposition 9.1 There exists a Borel set \( \mathcal{E} \subset {\mathbb{R}}^{N} \) of \( \mu \) -measure zero, such that \( {D}_{\mu }^{ \pm }\nu \) is finite, and \( {D}_{\mu }\nu = {D}_{\mu }^{ \pm }\nu \) in \( {\mathbb{R}}^{N} - \mathcal{E} \) . | Proof Assume first that both \( \mu \) and \( \nu \) are finite and set \( {\mathcal{E}}_{\infty } = \left\lbrack {{D}_{\mu }^{ + }\nu = \infty }\right\rbrack \) . By (8.2)\n\n\[{\mu }_{e}\left( \left\lbrack {{D}_{\mu }^{ + }\nu > t}\right\rbrack \right) \leq \frac{1}{t}\nu \left( {\mathbb{R}}^{N}\right) \;\text{ for a... | Yes |
For all fixed \( \rho > 0 \), the two functions\n\n\[ \n{\mathbb{R}}^{N} \ni x \rightarrow \mu \left( {{B}_{\rho }\left( x\right) }\right) ,\;{\mathbb{R}}^{N} \ni x \rightarrow \nu \left( {{B}_{\rho }\left( x\right) }\right) \n\]\n\nare upper semi-continuous. | The statement for \( x \rightarrow \mu \left( {{B}_{\rho }\left( x\right) }\right) \) reduces to\n\n\[ \n\mathop{\limsup }\limits_{{y \rightarrow x}}\mu \left( {{B}_{\rho }\left( y\right) }\right) \leq \mu \left( {{B}_{\rho }\left( x\right) }\right) \;\text{ for all }x \in {\mathbb{R}}^{N}. \n\]\n\nLet \( \left\{ {x}_{... | Yes |
Lemma 10.1 Let \( \nu \ll \mu \) . Then \( \nu \left( \left\lbrack {{D}_{\mu }\nu = 0}\right\rbrack \right) = 0 \) . | Proof Let \( \mathcal{E} \) be the Borel set claimed by Proposition 9.1 and appearing in (9.1). Then for all \( t > 0,\left\lbrack {{D}_{\mu }\nu = 0}\right\rbrack \subset \mathcal{E} \cup \left\lbrack {{D}_{\mu }^{ - }\nu < t}\right\rbrack \) . From this,(8.3), and the absolute continuity of \( \nu \) with respect to ... | Yes |
Proposition 10.1 Assume \( \nu \) is absolutely continuous with respect to \( \mu \) . Then for every \( \mu \) -measurable set \( E \)\n\n\[ \nu \left( E\right) = {\int }_{E}{D}_{\mu }{\nu d\mu } \] | Proof Let \( E \subset {\mathbb{R}}^{N} \) be \( \mu \) -measurable and for \( t > 1 \) and \( n \in \mathbb{Z} \) set\n\n\[ {E}_{n} = E \cap \left\lbrack {{t}^{n} < {D}_{\mu }\nu \leq {t}^{n + 1}}\right\rbrack . \]\n\nBy construction \( \mathop{\bigcup }\limits_{{n \in \mathbb{Z}}}{E}_{n} \subset E \), and\n\n\[ E - \... | Yes |
Proposition 10.2 Assume \( \nu \) is singular with respect to \( \mu \) . There exists a Borel set \( {\mathcal{E}}_{ \bot } \) of \( \mu \) -measure zero, such that \( {D}_{\mu }\nu = 0 \) in \( {\mathbb{R}}^{N} - {\mathcal{E}}_{ \bot } \) . | Proof Since \( \mu \) and \( \nu \) are singular, \( {\mathbb{R}}^{N} \) can be partitioned into two disjoint sets \( {\mathbb{R}}_{\mu }^{N} \) and \( {\mathbb{R}}_{\nu }^{N} \), such that for every \( E \in \mathcal{A} \) (§ 18 of Chap. 4).\n\n\[ \mu \left( {E \cap {\mathbb{R}}_{\nu }^{N}}\right) = 0\;\text{ and }\;\... | Yes |
Theorem 11.2 Let \( \mu \) be a Radon measure in \( {\mathbb{R}}^{N} \) and let \( f \) be locally \( \mu \) -integrable. There exists a \( \mu \) -measurable set \( \mathcal{E} \subset {\mathbb{R}}^{N} \) of \( \mu \) -measure zero, such that (11.2) holds for all \( x \in {\mathbb{R}}^{N} - \mathcal{E} \) . Equivalent... | Proof Let \( {r}_{n} \) be a rational number. The function \( \left| {f - {r}_{n}}\right| \) is locally \( \mu \) -integrable. Therefore there exists a Borel set \( {\mathcal{E}}_{n} \subset {\mathbb{R}}^{N} \) of \( \mu \) -measure zero, such that\n\n\[ \mathop{\lim }\limits_{{\rho \rightarrow 0}}\frac{1}{\mu \left( {... | Yes |
Proposition 12.1 Let \( f \) locally \( \mu \) -integrable in \( {\mathbb{R}}^{N} \) . Then if \( x \) is a Lebesgue point for \( f \) and \( {\mathcal{F}}_{x} \) is a regular family at \( x \)\n\n\[ \mathop{\lim }\limits_{\substack{{diam} \\ {S \in {\mathcal{F}}_{x}} }}\frac{1}{\mu \left( S\right) }{\int }_{S}\left| {... | Proof Having fixed \( S \in {\mathcal{F}}_{x} \) let \( B\left( x\right) \) be the ball satisfying (12.1). Then\n\n\[ \frac{1}{\mu \left( S\right) }{\int }_{S}\left| {f - f\left( x\right) }\right| {d\mu } \leq \frac{c}{\mu \left( {B\left( x\right) }\right) }{\int }_{B\left( x\right) }\left| {f - f\left( x\right) }\righ... | Yes |
Proposition 13.1 Let \( \\left\\{ {f}_{\\alpha }\\right\\} \) be a family of convex functions defined in \( \\left( {a, b}\\right) \) . Then the function \( f = \\sup {f}_{\\alpha } \) is convex in \( \\left( {a, b}\\right) \) . | Proof Fix \( x, y \\in \\left( {a, b}\\right) \) and \( t \\in \\left\\lbrack {0,1}\\right\\rbrack \) and assume first that \( f\\left( {{tx} + \\left( {1 - t}\\right) y}\\right) \) is finite. Having fixed an arbitrary \( \\varepsilon > 0 \) there exists \( \\alpha \) such that\n\n\[ f\\left( {{tx} + \\left( {1 - t}\\r... | Yes |
Proposition 13.2 Let \( f \) be a real-valued, convex function in some interval \( \left( {a, b}\right) \subset \) \( \mathbb{R} \) . Then the function\n\n\[ \n y \rightarrow \mathcal{F}\left( {x;y}\right) = \frac{f\left( x\right) - f\left( y\right) }{x - y}\;x, y \in \left( {a, b}\right) ;\;x \neq y \n\]\n\nis non-dec... | Proof Assume \( y > x \) . It suffices to show that\n\n\[ \n\mathcal{F}\left( {x;z}\right) \leq \mathcal{F}\left( {x;y}\right) \;\text{ for }\;z = {tx} + \left( {1 - t}\right) y\;\text{ for all }\;t \in \left( {0,1}\right) .\n\]\n\nBy the convexity of \( f \)\n\n\[ \n\mathcal{F}\left( {x;z}\right) = \frac{f\left( {{tx}... | Yes |
Proposition 13.3 Let \( f \) be a real-valued, convex function in some interval \( \left\lbrack {a, b}\right\rbrack \subset \) \( \mathbb{R} \) . Then \( f \) is locally Lipschitz continuous in \( \left( {a, b}\right) \) . | Proof Fix a subinterval \( \left\lbrack {c, d}\right\rbrack \subset \left( {a, b}\right) \) . Then for all \( x, y \in \left\lbrack {c, d}\right\rbrack \)\n\n\[ \mathcal{F}\left( {c;a}\right) = \frac{f\left( c\right) - f\left( a\right) }{c - a} \leq \frac{f\left( x\right) - f\left( y\right) }{x - y} \leq \frac{f\left( ... | Yes |
Proposition 13.4 Let \( f \) be a real-valued convex function in \( \left( {a, b}\right) \) . Then \( f \) is a.e. differentiable in \( \left( {a, b}\right) \) . Moreover the right and left derivatives \( {D}_{ \pm }f\left( x\right) \) exist and are finite at each \( x \in \left( {a, b}\right) \) and are both monotone ... | Proof For each \( x \in \left( {a, b}\right) \) fixed, the function \( \left( {h, k}\right) \rightarrow \mathcal{F}\left( {x + h;x + k}\right) \) is nondecreasing in both variables. Therefore there exist and are finite the limits\n\n\[ \n{D}_{ - }f\left( x\right) = \mathop{\lim }\limits_{{h \nearrow 0}}\mathcal{F}\left... | Yes |
Proposition 14.1 (Jensen [76]) Let \( E \) be a measurable set of finite measure, and let \( f : E \rightarrow \mathbb{R} \) be integrable in \( E \) . Then, for every real-valued, convex function \( \varphi \) defined in \( \mathbb{R} \)\n\n\[ \varphi \left( {\frac{1}{\mu \left( E\right) }{\int }_{E}{fd\mu }}\right) \... | Proof Applying (14.1) for the choices\n\n\[ \alpha = \frac{1}{\mu \left( E\right) }{\int }_{E}{fd\mu }\;\eta = f\left( x\right) \;\text{ for a.e. }x \in E \]\n\nyields\n\n\[ \varphi \left( {\frac{1}{\mu \left( E\right) }{\int }_{E}{fd\mu }}\right) + m\left( {f\left( x\right) - \frac{1}{\mu \left( E\right) }{\int }_{E}{... | Yes |
Theorem 15.1 (Kirzbraun-McShane-Pucci) \( {}^{3} \) Let \( f \) be a real-valued, uniformly continuous function on a set \( E \subset {\mathbb{R}}^{N} \) with modulus of continuity \( {\omega }_{f} \) satisfying (15.1). There exists a continuous function \( \widetilde{f} \) defined on \( {\mathbb{R}}^{N} \), which coin... | Proof For each \( x \in {\mathbb{R}}^{N} \), set\n\n\[ g\left( x\right) = \mathop{\inf }\limits_{{y \in E}}\left\{ {f\left( y\right) + {c}_{f}\left( \left| {x - y}\right| \right) }\right\} .\n\]\n\nThe required extension is\n\n\[ \widetilde{f}\left( x\right) = \min \left\{ {g\left( x\right) ;\mathop{\sup }\limits_{E}f}... | Yes |
Corollary 16.1 Let \( E \) be a compact subset of \( {\mathbb{R}}^{N} \). Then \( C\left( E\right) \) endowed with the topology of the uniform convergence, is separable. | Proof The collection of polynomials in the real variables \( {x}_{1},\ldots ,{x}_{N} \), with rational coefficients is a countable, dense subset of \( C\left( E\right) \) . | Yes |
Proposition 18.1 Let \( \{ X;\mathcal{U}\} \) be a compact Hausdorff space and let \( \mathcal{F} \subset C\left( X\right) \) be an algebra. Then\n\n(i) The closure \( \overline{\mathcal{F}} \) in \( C\left( X\right) \), is an algebra\n\n(ii) If \( f \in \mathcal{F} \) then \( \left| f\right| \in \overline{\mathcal{F}}... | Proof The first statement follows from the structure of an algebra and the notion of closure in the metric (17.1). To prove (ii) we may assume, without loss of generality, that \( \left| f\right| \leq 1 \) . Regard \( f \) as a variable ranging over \( \left\lbrack {-1,1}\right\rbrack \) . By the classical Weierstrass ... | Yes |
Proposition 19.1 A subset \( \mathcal{K} \subset C\left( \bar{E}\right) \) is pre-compact in \( C\left( \bar{E}\right) \), if and only if the elements of \( K \) are pointwise equibounded and equi-continuous in \( \bar{E} \) . | Proof Since \( C\left( E\right) \) is metric and separable, compactness and sequential compactness are equivalent (Proposition 17.3 of Chap. 2). Then, the sufficient condition follows from Theorem 19.1. For the necessary part recall that if \( \mathcal{K} \) is pre-compact in \( C\left( \bar{E}\right) \) , then \( \ove... | No |
Proposition 2.1 (Young’s Inequality) \( {}^{1} \) Let \( 1 \leq p, q \leq \infty \) be conjugate. Then for all \( a, b \in \mathbb{R} \)\n\n\[ \left| {ab}\right| \leq \frac{1}{p}{\left| a\right| }^{p} + \frac{1}{q}{\left| b\right| }^{q} \]\n\n(2.2)\n\nand equality holds only if \( {\left| a\right| }^{p} = {\left| b\rig... | Proof The inequality is obvious if either \( a \) or \( b \) is zero. Thus assume \( \left| a\right| > 0 \) and \( \left| b\right| > 0 \) . The inequality is also obvious if either \( p = 1 \) or \( q = 1 \) . Thus assume \( 1 < p, q < \infty \) . The function\n\n\[ s \rightarrow \left( {\frac{{s}^{p}}{p} + \frac{1}{q}... | Yes |
Proposition 2.2 (Hölder’s Inequality) Let \( f \in {L}^{p}\left( E\right) \) and \( g \in {L}^{q}\left( E\right) \), where \( 1 \leq \) \( p, q \leq \infty \) satisfy (2.1). Then \( {fg} \in {L}^{1}\left( E\right) \) and \[ {\int }_{E}\left| {fg}\right| {d\mu } \leq \parallel f{\parallel }_{p}\parallel g{\parallel }_{q... | Proof May assume that \( f \) and \( g \) are nonnegative and neither is zero a.e. in \( E \) . Also (2.3) is obvious if either \( p = 1 \) or \( q = 1 \) . If \( p, q > 1 \), in (2.2) take \[ a = \frac{f}{\parallel f{\parallel }_{p}}\;\text{ and }\;b = \frac{g}{\parallel g{\parallel }_{q}} \] to obtain \[ \frac{fg}{\p... | Yes |
Proposition 2.3 (Minkowski Inequality) Let \( f, g \in {L}^{p}\left( E\right) \) for some \( 1 \leq p \leq \infty \) . Then\n\n\[ \parallel f + g{\parallel }_{p} \leq \parallel f{\parallel }_{p} + \parallel g{\parallel }_{p} \] | Proof The inequality is obvious if \( p = 1 \) and \( p = \infty \) . If \( 1 < p < \infty \)\n\n\[ \parallel f + g{\parallel }_{p}^{p} = {\int }_{E}{\left| f + g\right| }^{p}{d\mu } = {\int }_{E}{\left| f + g\right| }^{p - 1}\left| {f + g}\right| {d\mu } \]\n\n\[ \leq {\int }_{E}{\left| f + g\right| }^{p - 1}\left| f\... | Yes |
Proposition 3.1 Let \( f \in {L}^{p}\left( E\right) \) for some \( 1 \leq p < \infty \) . Then\n\n\[ \parallel f{\parallel }_{p} = {\left( {\int }_{E}{\left| f\right| }^{p}d\mu \right) }^{1/p} = \mathop{\sup }\limits_{\substack{{g \in {L}^{q}\left( E\right) } \\ {\parallel g{\parallel }_{q} = 1} }}{\int }_{E}{fgd\mu } ... | Proof May assume that \( f ≢ 0 \) . By Hölder’s inequality\n\n\[ \mathop{\sup }\limits_{\substack{{g \in {L}^{q}\left( E\right) } \\ {\parallel g{\parallel }_{q} = 1} }}{\int }_{E}{fgd\mu } \leq \parallel f{\parallel }_{p} \]\n\nIf \( 1 < p < \infty \) one verifies that\n\n\[ {g}_{ * } = \frac{{\left| f\right| }^{p - 2... | Yes |
Proposition 3.2 Let \( \\mu \\left( E\\right) < \\infty \) and \( f \\in {L}^{\\infty }\\left( E\\right) \) . Then\n\n\[ \n\\mathop{\\lim }\\limits_{{p \\rightarrow \\infty }}\\parallel f{\\parallel }_{p} = \\parallel f{\\parallel }_{\\infty }\n\] | Proof Since \( E \) is of finite measure\n\n\[ \n\\mathop{\\limsup }\\limits_{{p \\rightarrow \\infty }}\\parallel f{\\parallel }_{p} \\leq \\parallel f{\\parallel }_{\\infty }\\mathop{\\limsup }\\limits_{{p \\rightarrow \\infty }}\\mu {\\left( E\\right) }^{1/p} = \\parallel f{\\parallel }_{\\infty }.\n\]\n\nNext, for ... | Yes |
Proposition 3.3 Let \( \{ X,\mathcal{A},\mu \} \) and \( \{ Y,\mathcal{B},\nu \} \) be two complete measure spaces and assume in addition that \( \{ Y,\mathcal{B},\nu \} \) is \( \sigma \) -finite. Then for every nonnegative \( f \in {L}^{p}(X \times \) Y) for some \( 1 \leq p < \infty \)\n\n\[{\left( {\int }_{X}{\left... | Proof Assume first that \( \nu \left( Y\right) < \infty \) . Then for every \( g \in {L}^{q}\left( X\right) \) one has \( {fg} \in {L}^{1}(X \times \) \( Y) \) . Setting\n\n\[F = {\int }_{Y}f\left( {\cdot, y}\right) {d\nu }\]\n\nthe left hand side is \( \parallel F{\parallel }_{p, X} \) . Then\n\n\[\parallel F{\paralle... | Yes |
Proposition 6.1 Let \( f \in {L}^{p}\left( E\right) \) for some \( 1 \leq p \leq \infty \) . For every \( \varepsilon > 0 \) there exists a simple function \( \varphi \in {L}^{p}\left( E\right) \), such that \( \parallel f - \varphi {\parallel }_{p} \leq \varepsilon \). | Proof By the decomposition \( f = {f}^{ + } - {f}^{ - } \), one may assume that \( f \) is nonnegative. Since \( f \) is measurable, there exists a sequence \( \left\{ {\varphi }_{n}\right\} \) of nonnegative, simple functions such that\n\n\[{\varphi }_{n} \leq {\varphi }_{n + 1}\;\text{ and }\;{\varphi }_{n} \rightarr... | No |
Proposition 6.2 Let \( 1 \leq p, q \leq \infty \) be conjugate, and let \( g \in {L}^{1}\left( E\right) \) satisfy\n\n\[{\int }_{E}{\varphi gd\mu } \leq K\parallel \varphi {\parallel }_{p}\text{ for all simple functions }\varphi\]\n\nfor some positive constant \( K \) . Then \( g \in {L}^{q}\left( E\right) \) and \( \p... | Proof If \( q = 1 \) it suffices to choose \( \varphi = \operatorname{sign}g \in {L}^{\infty }\left( E\right) \) . Assuming \( q \in \left( {1,\infty }\right) \), let \( \left\{ {\varphi }_{n}\right\} \) denote a sequence on nonnegative simple functions, such that \( {\varphi }_{n} \leq {\varphi }_{n + 1} \) and \( {\v... | Yes |
Proposition 8.1 Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of functions in \( {L}^{p}\\left( E\\right) \) for some \( 1 \\leq p < \\infty \) , converging weakly to some \( f \\in {L}^{p}\\left( E\\right) \). Then\n\n\[ \n\\liminf {\\begin{Vmatrix}{f}_{n}\\end{Vmatrix}}_{p} \\geq \\parallel f{\\parallel }_{p} \... | Proof Assume first that \( 1 \\leq p < \\infty \). The function \( g = {\\left| f\\right| }^{p/q}\\operatorname{sign}f \) belongs to \( {L}^{q}\\left( E\\right) \)\n\nand\n\n\[ \n\\lim {\\int }_{E}{f}_{n}{gd\\mu } = {\\int }_{E}{fgd\\mu } = \\parallel f{\\parallel }_{p}^{p}.\n\]\n\nOn the other hand, by Hölder's inequa... | Yes |
Lemma 9.2 Let \( 1 < p < 2 \) . There exists a constant \( c \in \left( {0,1}\right) \) such that for all \( t \in \mathbb{R} \)\n\n\[{\left| 1 + t\right| }^{p} \geq \left\{ \begin{array}{lll} 1 + {pt} + c{\left| t\right| }^{p} & \text{ if } & \left| t\right| \geq 1 \\ 1 + {pt} + c{\left| t\right| }^{2} & \text{ if } &... | The proof of these lemmas is given in \( §{9.1}\mathrm{c} \) of the Complements. | No |
Proposition 10.1 Let \( 1 < p \leq \infty \) and \( 1 \leq q < \infty \) be conjugate. Every \( g \in {L}^{q}\left( E\right) \) generates a bounded linear functional in \( {L}^{p}\left( E\right) \) by the formula\n\n\[ \n{\mathcal{F}}_{g}\left( f\right) = {\int }_{E}{fgd\mu }\;\text{ for all }f \in {L}^{p}\left( E\righ... | Proof The map \( {\mathcal{F}}_{g} \) is linear. By Hölder’s inequality it is also bounded. If \( 1 \leq q < \infty \) , Proposition 3.1 identifies the norm \( \begin{Vmatrix}{\mathcal{F}}_{g}\end{Vmatrix} \) as the norm \( \parallel g{\parallel }_{q} \) .\n\nLet now \( q = \infty \) and assume momentarily that \( E \)... | Yes |
Lemma 12.1 Let \( 1 < p, q < \infty \) be conjugate. For every fixed \( t > 0 \) there holds\n\n\[ \varphi \left( {s;t}\right) \leq {\left| 1 + t\right| }^{p} + {\left| 1 - t\right| }^{p}\text{ for }p \in (1,2\rbrack \]\n\n\[ \varphi \left( {s;t}\right) \geq {\left| 1 + t\right| }^{p} + {\left| 1 - t\right| }^{p}\text{... | Proof Assume first \( t \in \left( {0,1}\right) \) . By direct calculation\n\n\[ \frac{1}{p - 1}\frac{{d\varphi }\left( {s;t}\right) }{ds} = \left\lbrack {{\left( 1 + s\right) }^{p - 2} - {\left( 1 - s\right) }^{p - 2}}\right\rbrack \frac{{s}^{p} - {t}^{p}}{{s}^{p}}. \]\n\nTherefore if \( p \in \left( {1,2}\right) \) t... | Yes |
Proposition 13.1 The spaces \( {L}^{p}\left( E\right) \) for \( 1 < p < \infty \) are uniformly convex. | Proof It suffices to verify (4.4). Let \( f, g \in {L}^{p}\left( E\right) \) satisfy \( \parallel f{\parallel }_{p} = \parallel g{\parallel }_{p} = 1 \), and \( \parallel f - g{\parallel }_{p} \geq \varepsilon > 0 \) . By the Clarkson inequalities\n\n\[{\begin{Vmatrix}\frac{f + g}{2}\end{Vmatrix}}_{p}^{p} \leq 1 - \fra... | Yes |
Proposition 13.2 Let \( 1 < p, q < \infty \) be conjugate. For a nonzero \( g \in {L}^{q}\left( E\right) \) let \( {g}^{ * } \) be defined by (10.4).\n\nIf \( {\mathcal{F}}_{1} \) and \( {\mathcal{F}}_{2} \) are two bounded linear functionals in \( {L}^{p}\left( E\right) \) satisfying\n\n\[ \n{\mathcal{F}}_{i}\left( {g... | Proof If \( {\mathcal{F}}_{1} ≢ {\mathcal{F}}_{2} \), there exists \( f \in {L}^{p}\left( E\right) \) such that \( {\mathcal{F}}_{1}\left( f\right) \neq {\mathcal{F}}_{2}\left( f\right) \) . Set\n\n\[ \n\varphi = \frac{2f}{{\mathcal{F}}_{1}\left( f\right) - {\mathcal{F}}_{2}\left( f\right) } - \frac{{\mathcal{F}}_{1}\l... | Yes |
Theorem 14.1 Let \( 1 < p, q < \infty \) be conjugate. To every bounded, linear functional \( \mathcal{F} \) in \( {L}^{p}\left( E\right) \), there corresponds a unique function \( g \in {L}^{q}\left( E\right) \) such that \( \mathcal{F} \) is represented by the formula (10.2). Moreover \( \parallel \mathcal{F}\paralle... | ## 14.1 Proof of Theorem 14.1. The Case \( 1 < p < \infty \)\n\nWithout loss of generality we may assume \( \parallel \mathcal{F}\parallel = 1 \) . By the definition (10.1) of \( \parallel \mathcal{F}\parallel \), there exists a sequence \( \left\{ {f}_{n}\right\} \) of functions in \( {L}^{p}\left( E\right) \), such t... | Yes |
Proposition 15.1 Let \( E \subset {\mathbb{R}}^{N} \) be Lebesgue measurable and of positive measure. Then \( {L}^{\infty }\left( E\right) \) is not separable, i.e., it does not contain a dense sequence \( \left\{ {f}_{n}\right\} \) . | Proof Denote by \( \left\{ {{B}_{s}\left( y\right) }\right\} \) the collection of open balls of radius \( s \) centered at \( y \) and such that \( \mu \left( {E \cap {B}_{s}\left( y\right) }\right) > 0 \) . For each \( s > 0 \) fixed there exists uncountably many \( r \) such that\n\n\[ \n{\begin{Vmatrix}{\chi }_{E \c... | Yes |
Proposition 18.1 Let \( f \in {L}^{p}\left( E\right) \) for some \( 1 \leq p < \infty \) . Then\n\n\[ \n\left( {{J}_{\varepsilon } * f}\right) \in {L}^{p}\left( E\right) \;\text{ and }\;{\begin{Vmatrix}\left( {J}_{\varepsilon } * f\right) \end{Vmatrix}}_{p} \leq \parallel f{\parallel }_{p} \n\]\n\nand\n\n\[ \n\mathop{\... | Proof By Hölder's inequality\n\n\[ \n\left| {\left( {{J}_{\varepsilon } * f}\right) \left( x\right) }\right| = \left| {{\int }_{{\mathbb{R}}^{N}}{J}_{\varepsilon }\left( {x - y}\right) f\left( y\right) {dy}}\right| \n\]\n\n\[ \n\leq {\left( {\int }_{{\mathbb{R}}^{N}}{J}_{\varepsilon }\left( x - y\right) dy\right) }^{1/... | Yes |
Proposition 18.2 Let \( E \) be an open subset of \( {\mathbb{R}}^{N} \) . Then \( {C}_{o}^{\infty }\left( E\right) \) is dense in \( {L}^{p}\left( E\right) \) for \( p \in \lbrack 1,\infty ) \) . | Proof Having fixed \( \varepsilon > 0 \), let \( {K}_{\varepsilon } \) be a compact subset of \( E \) such that\n\n\[{\int }_{E - {K}_{\varepsilon }}{\left| f\right| }^{p}{dy} \leq \frac{1}{{2}^{p}}{\varepsilon }^{p}\]\n\nLet \( 2{\delta }_{\varepsilon } = \operatorname{dist}\left\{ {{K}_{\varepsilon };\partial E}\righ... | Yes |
Proposition 2.1 There exists no function \( f \in X \) such that and\n\n\[ \parallel f\parallel = 1\;\text{ and }\;\parallel f - \mathbf{g}\parallel \geq 1\;\text{ for all functions }\mathbf{g} \in {X}_{o}. \]\n\n(2.2) | Proof (of Proposition 2.1) Assume such a \( f \) exists. For \( h \in X - {X}_{o} \) set\n\n\[ g = f - {ch}\;\text{ where }\;c = \frac{{\int }_{0}^{1}f\left( t\right) {dt}}{{\int }_{0}^{1}h\left( t\right) {dt}}. \]\n\nThen \( \mathbf{g} \in {X}_{o} \) and\n\n\[ 1 \leq \parallel f - \left( {f - {ch}}\right) \parallel = ... | Yes |
Lemma 2.1 (Riesz [128]) Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space and let \( \left\{ {{X}_{o};\parallel \cdot \parallel }\right\} \) be a closed, proper subspace of \( \{ X;\parallel \cdot \parallel \} \). For every \( \varepsilon \in \left( {0,1}\right) \) there exists \( {x}_{\varepsilon } \in X \... | Proof Fix \( {x}_{o} \in X - {X}_{o} \) and let \( d \) be the distance from \( {x}_{o} \) to \( {X}_{o} \). Since \( {X}_{o} \) is a proper, closed subspace of \( X \) we have \( d > 0 \). There exists an element \( {x}_{1} \in {X}_{o} \), such that\n\n\[ d \leq \begin{Vmatrix}{{x}_{o} - {x}_{1}}\end{Vmatrix} \leq d +... | Yes |
Proposition 2.2 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space. If the unit sphere \( {S}_{1} \) is compact, then \( X \) is finite dimensional. | Proof If not, choose \( {x}_{1} \in {S}_{1} \) and consider the subspace \( {X}_{1} \) spanned by \( {x}_{1} \) . It is a proper subspace of \( X \) and by Corollary 12.1 of Chap. 2, it is closed. Therefore by the Riesz lemma, there exists \( {x}_{2} \in {S}_{1} \) such that \( \begin{Vmatrix}{{x}_{1} - {x}_{2}}\end{Vm... | Yes |
Proposition 3.1 Let \( T \) be a linear map from a normed space \( \\left\\{ {X;\\parallel \\cdot {\\parallel }_{X}}\\right\\} \) into a normed space \( \\left\\{ {Y;\\parallel \\cdot {\\parallel }_{Y}}\\right\\} \) . Then\n\n(i). If \( T \) is bounded, it is uniformly continuous.\n\n(ii). If \( T \) is continuous at s... | Proof If \( T \) is bounded, for any pair of elements \( x, y \\in X \)\n\n\[ \n\\parallel T\\left( x\\right) - T\\left( y\\right) {\\parallel }_{Y} \\leq \\parallel T\\parallel \\parallel x - y{\\parallel }_{X} \n\]\n\nThis proves (i). To prove (ii) we may assume that \( {x}_{o} = \\Theta \) . There exists a ball \( {... | Yes |
Proposition 3.2 Let \( \left\{ {Y;\parallel \cdot {\parallel }_{Y}}\right\} \) be a Banach space. Then \( \mathcal{B}\left( {X;Y}\right) \) endowed with the norm (3.1) is also a Banach space. | Proof Let \( \left\{ {T}_{n}\right\} \) be a Cauchy sequence in \( \mathcal{B}\left( {X;Y}\right) \), i.e., for any fixed \( \varepsilon > 0 \), there exist an index \( {n}_{\varepsilon } \) such that, \( \begin{Vmatrix}{{T}_{n} - {T}_{m}}\end{Vmatrix} \leq \varepsilon \) for all \( n, m \geq {n}_{\varepsilon } \) . Fo... | Yes |
Proposition 5.1 Let \( {X}_{o} \) be a closed subspace of \( X \) such that the quotient space \( X/{X}_{o} \) is 1 -dimensional. Then, there exists a nontrivial, bounded, linear functional \( T : X \rightarrow \mathbb{R} \) such that \( {X}_{o} = \ker \{ T\} \) . | Proof To prove the first statement choose \( x \in X - {X}_{o} \) such that \( \operatorname{dist}\left\{ {x;{X}_{o}}\right\} > 0 \) and write \( X - {X}_{o} = \bigcup \{ {\lambda x} \mid \lambda \in \mathbb{R}\} \) . Every element \( y \in X \) can be written as \( y = {x}_{o} + {\lambda x} \) for some \( {x}_{o} \in ... | No |
Corollary 5.2 Let \( T \) be a not identically zero linear functional on a normed space \( \{ X;\parallel \cdot \parallel \} \) . Then \( T \) is continuous if and only if \( \ker \{ T\} \) is not dense in \( X \) . | As a consequence \( T \) is continuous if and only if \( \ker \{ T\} \) is nowhere dense in \( X \) . | No |
Proposition 6.1 Let \( \mathcal{T} \) be a family of bounded linear maps from a Banach space \( \left\{ {X;\parallel \cdot {\parallel }_{X}}\right\} \) into a normed space \( \left\{ {Y;\parallel \cdot {\parallel }_{Y}}\right\} \) . Assume that the elements of \( \mathcal{T} \) are pointwise equibounded in \( X \), i.e... | Proof The functions \( \parallel T\left( x\right) {\parallel }_{Y} : X \rightarrow \mathbb{R} \) satisfy the assumptions of the Banach-Steinhaus theorem. Therefore there exists a positive \( F \) and a ball \( {B}_{\varepsilon }\left( {x}_{o}\right) \subset X \), of radius \( \varepsilon \) and centered at some \( {x}_... | Yes |
Corollary 6.1 Let \( \{ X;\parallel \cdot \parallel \} \) be a Banach space. Then a pointwise bounded family of elements in \( {X}^{ * } \) is equibounded in \( X \) . | ## 6.1 Another Proof of Proposition 6.1\n\nThe proposition can be proved without appealing to category arguments. It suffices to establish that the elements of \( \mathcal{T} \) are equiuniformly bounded in some open ball \( {B}_{\varepsilon }\left( {x}_{o}\right) \subset X \) . The proof proceeds by contradiction, ass... | Yes |
Theorem 7.1 ([Banach [13]) Let \( T \) be a contraction form a Banach space \( \{ X;\parallel \cdot \parallel \} \) into itself. Then \( T \) has a unique fixed point, i.e., there exists a unique \( {x}_{o} \in X \) such that \( T\left( {x}_{o}\right) = {x}_{o} \) . | Proof Starting from an arbitrary \( {x}_{1} \in X \), define the sequence \( {x}_{n + 1} = T\left( {x}_{n}\right) \) . Then\n\n\[ \begin{Vmatrix}{T\left( {x}_{n + 1}\right) - T\left( {x}_{n}\right) }\end{Vmatrix} \leq {t}^{n}\begin{Vmatrix}{{x}_{2} - {x}_{1}}\end{Vmatrix}. \]\n\nEquivalently\n\n\[ \begin{Vmatrix}{T\lef... | Yes |
Proposition 7.1 Assume the constant \( \gamma \) in (4.3) is less than 1 . Then the integral equation (7.1) has a unique solution. | Proof The solution would be the unique fixed point of\n\n\[ T\left( f\right) = {\int }_{E}K\left( {x, y}\right) f\left( y\right) {dy} + h \]\n\nprovided \( T : {L}^{1}\left( E\right) \rightarrow {L}^{1}\left( E\right) \) is a contraction. For \( f,\mathbf{g} \in {L}^{1}\left( E\right) \)\n\n\[ \parallel T\left( {f - \m... | Yes |
Lemma 8.1 \( T\left( {B}_{1}\right) \) contains a ball \( {\mathcal{B}}_{\varepsilon } \) for some \( \varepsilon > 0 \) . | Proof Since \( T \) is linear and onto\n\n\[X = \bigcup n{B}_{1/2}\;\text{ and }\;Y = \bigcup {nT}\left( {B}_{1/2}\right) .\n\]\n\nSince \( \left\{ {Y;\parallel \cdot {\parallel }_{Y}}\right\} \) is of second category, \( T\left( {B}_{1/2}\right) \) is not nowhere dense and its closure contains an open ball \( {\mathca... | Yes |
Proposition 8.1 Let \( \parallel \cdot {\parallel }_{1} \) and \( \parallel \cdot {\parallel }_{2} \) be two norms on the same vector space \( X \), by which \( \left\{ {X;\parallel \cdot {\parallel }_{1}}\right\} \) and \( \left\{ {X;\parallel \cdot {\parallel }_{2}}\right\} \) are both Banach spaces. Assume that ther... | Proof The identity map \( T\left( x\right) = x \), from \( \left\{ {X;\parallel \cdot {\parallel }_{1}}\right\} \) onto \( \left\{ {X;\parallel \cdot {\parallel }_{2}}\right\} \) is linear and one-to-one. By (8.2) is also continuous. Therefore it is a homeomorphism. In particular the inverse \( {T}^{-1} \) is linear an... | Yes |
Theorem 8.2 (Closed Graph Theorem) Let \( T \) be a linear map from a Banach space \( \\left\\{ {X;\\parallel \\cdot {\\parallel }_{X}}\\right\\} \) into a Banach space \( \\left\\{ {Y;\\parallel \\cdot {\\parallel }_{Y}}\\right\\} \) . If \( {\\mathcal{G}}_{T} \) is closed, then \( T \) is continuous. | Proof On \( X \) introduce a new norm \( \\parallel x\\parallel = \\parallel x{\\parallel }_{X} + \\parallel T\\left( x\\right) {\\parallel }_{Y} \) . One verifies this is a norm on \( X \) and if \( {\\mathcal{G}}_{T} \) is closed, \( \\{ X;\\parallel \\cdot \\parallel \\} \) is complete. Therefore if \( {\\mathcal{G}... | Yes |
Proposition 10.1 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space and let \( {X}_{o} \) be a subspace of \( X \) . Every \( {T}_{o} \in {X}_{o}^{ * } \) admits an extension \( T \in {X}^{ * } \) such that \( \parallel T\parallel = \begin{Vmatrix}{T}_{o}\end{Vmatrix} \) . | Proof Apply the Hahn-Banach theorem with \( p\left( x\right) = \begin{Vmatrix}{T}_{o}\end{Vmatrix}\parallel x\parallel \) . This gives an extension \( T \) defined in \( X \) and satisfying\n\n\[ \left| {T\left( x\right) }\right| \leq \begin{Vmatrix}{T}_{o}\end{Vmatrix}\parallel x\parallel \;\text{ for all }x \in X. \]... | Yes |
Proposition 10.2 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space. For every \( {x}_{o} \in X \), and \( {x}_{o} \neq \Theta \) , there exists \( T \in {X}^{ * } \) such that \( \parallel T\parallel = 1 \) and \( T\left( {x}_{o}\right) = \begin{Vmatrix}{x}_{o}\end{Vmatrix} \) . | Proof Having fixed \( {x}_{o} \in X \) let \( {X}_{o} = \left\{ {\lambda {x}_{o} \mid \lambda \in \mathbb{R}}\right\} \) be the span of \( {x}_{o} \) . On \( {X}_{o} \) consider the functional \( {T}_{o}\left( {\lambda {x}_{o}}\right) = \lambda \begin{Vmatrix}{x}_{o}\end{Vmatrix} \) and as \( p \) take the norm \( \par... | Yes |
Corollary 10.1 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space. Then \( {X}^{ * } \) separates the points of \( X \), i.e., for any pair \( x, y \) of distinct points of \( X \), there exists \( T \in {X}^{ * } \) such that \( T\left( x\right) \neq T\left( y\right) \) . | Proof Apply Proposition 10.2 to the element \( x - y \) . | No |
Proposition 10.3 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space and let \( {X}_{o} \) be a linear subspace of \( X \) . Assume that there exists an element \( \eta \in X - {X}_{o} \) that has positive distance from \( {X}_{o} \), i.e., \[ \mathop{\inf }\limits_{{x \in {X}_{o}}}\parallel x - \eta \paralle... | Proof Let \( {X}_{\eta } \) be the span of \( {X}_{o} \) and \( \eta \) . On \( {X}_{\eta } \) define the linear functional \( {T}_{\eta }\left( {{\lambda \eta } + x}\right) = \) \( {\lambda \delta } \) for all \( x \in {X}_{o} \) and all \( \lambda \in \mathbb{R} \) . From the definition of \( \delta \) for \( \lambda... | Yes |
Proposition 10.4 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space. Then if \( {X}^{ * } \) is separable also \( \{ X;\parallel \cdot \parallel \} \) is separable. | Proof Let \( \left\{ {T}_{n}\right\} \) be a sequence dense in \( {X}^{ * } \) . For each \( {T}_{n} \) choose \( {x}_{n} \in X \) such that \( \begin{Vmatrix}{x}_{n}\end{Vmatrix} = 1 \) and \( \left| {{T}_{n}\left( {x}_{n}\right) }\right| \geq \frac{1}{2}\begin{Vmatrix}{T}_{n}\end{Vmatrix} \) . Let now \( {X}_{o} \) b... | Yes |
Proposition 11.1 The map \( x \rightarrow {\mu }_{C}\left( x\right) \) is sub-linear in \( X \) . | Proof Fix \( x, y \in X \) and let \( t \) and \( s \) be positive numbers such that \( {\mu }_{C}\left( x\right) < t \) and \( {\mu }_{C}\left( y\right) < s \) . For such choices, \( {t}^{-1}x \in C \) and \( {s}^{-1}y \in C \) . Then, since \( C \) is convex\n\n\[ \n\frac{1}{s + t}\left( {x + y}\right) = \frac{t}{s +... | Yes |
Proposition 11.2 Let \( {C}_{1} \) and \( {C}_{2} \) be two disjoint, convex subsets of a Hausdorff, linear, topological vector space \( \{ X;\mathcal{U}\} \), and assume \( {C}_{1} \) is open. There exists a nontrivial, linear, continuous functional \( T \) on \( X \), and \( \alpha \in \mathbb{R} \) such that\n\n\[ T... | Proof Fix some \( {x}_{1} \in {C}_{1} \) and \( {x}_{2} \in {C}_{2} \) and set\n\n\[ C = {C}_{1} - {C}_{2} + {x}_{o}\;\text{ where }{x}_{o} = {x}_{2} - {x}_{1}. \]\n\nThe set \( C \) is open, convex and it contains the origin. Since \( {C}_{1} \) and \( {C}_{2} \) are disjoint, \( {x}_{o} \notin C \) . On the one-dimen... | Yes |
Proposition 11.3 Let \( {C}_{1} \) and \( {C}_{2} \) be two nonempty, disjoint, convex subsets of a locally convex, Hausdorff, topological vector space \( \{ X;\mathcal{U}\} \) . Assume that \( {C}_{1} \) is compact and \( {C}_{2} \) is closed. There exists a bounded linear functional \( T \) on \( X \) and real number... | Proof We claim that there exists a convex, open neighborhood \( \mathcal{O} \) of the origin such that \( {C}_{1} + \mathcal{O} \) is open, convex and does not intersect \( {C}_{2} \) . Whence the claim is established, Proposition 11.3 follows from Proposition 11.2 applied to the pair of sets \( {C}_{1} + \mathcal{O} \... | Yes |
Proposition 12.1 Let \( X \) be infinite dimensional. Then every weak neighborhood of the origin, contains an infinite dimensional subspace \( {X}_{o} \) . | Proof Having fixed a weak neighborhood of the origin, we may assume is of the form (12.1)-(12.2) and set\n\n\[ \n{X}_{o} = \mathop{\bigcap }\limits_{{j = 1}}^{n}\ker \left\{ {T}_{j}\right\} \n\]\n\nThis is a subspace of \( X \) and \( {X}_{o} \subset \mathcal{O} \) . To prove that \( {X}_{o} \) is infinite dimensional,... | Yes |
Corollary 12.2 The weak topology of an infinite dimensional normed space does not satisfy the first axiom of countability. | Proof Assume by contradiction that \( \left\{ {\mathcal{O}}_{n}\right\} \) is a countable base for the weak topology at the origin. By Proposition 12.1 and Corollary 12.1 one may pick\n\n\[ \n{x}_{n} \in \mathop{\bigcap }\limits_{{j = 1}}^{n}{\mathcal{O}}_{j}\;\text{ such that }\begin{Vmatrix}{x}_{n}\end{Vmatrix} \geq ... | Yes |
Corollary 12.3 The weak topology of an infinite dimensional normed space is not metrizable, i.e., there is no metric \( d\left( {\cdot , \cdot }\right) \) on \( X \times X \) that generates its weak topology. | Proof If the weak topology of \( \{ X;\parallel \cdot \parallel \} \) were metrizable, it would satisfy the first axiom of countability. | No |
Proposition 12.2 (Mazur [105]) Let \( E \) be a convex subset of a normed space \( \{ X;\parallel \cdot \parallel \} \). Then, the weak closure of \( E \) coincides with its strong closure. | Proof Denote by \( {\bar{E}}_{w} \) and \( {\bar{E}}_{s} \) the closure of \( E \) in the weak and, respectively, strong topology. Since \( \mathcal{W} \) is weaker than the strong topology, \( {\bar{E}}_{s} \subset {\bar{E}}_{w} \). For the converse inclusion it suffices to show that \( X - {\bar{E}}_{s} \) is a weakl... | Yes |
Corollary 12.5 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space and let \( \left\{ {x}_{n}\right\} \) be a sequence of elements of \( X \) converging weakly to some \( x \in X \) . Then there exists a sequence \( \left\{ {y}_{m}\right\} \) of elements of \( X \), such that each \( {y}_{m} \) is the convex ... | Proof Let \( c\left( \left\{ {x}_{n}\right\} \right) \) be convex hull of \( \left\{ {x}_{n}\right\} \) and denote by \( {\overline{c\left( \left\{ {x}_{n}\right\} \right) }}_{w} \) its weak closure. By assumption the weak limit \( x \) belongs to \( {\overline{c\left( \left\{ {x}_{n}\right\} \right) }}_{w} \) . The co... | Yes |
Proposition 13.1 Let \( {X}_{o} \) be a closed, linear; proper subspace of a reflexive Banach space \( \{ X;\parallel \cdot \parallel \} \) . Then \( {X}_{o} \) is reflexive. | Proof By Proposition 10.1, every \( {x}_{o}^{ * } \in {X}_{o}^{ * } \) can be regarded as the restriction to \( {X}_{o} \) of some \( {x}^{ * } \in {X}^{ * } \), such that \( {\begin{Vmatrix}{x}^{ * }\end{Vmatrix}}_{{X}^{ * }} = {\begin{Vmatrix}{x}_{o}^{ * }\end{Vmatrix}}_{{X}_{o}^{ * }} \) . Now fix \( {f}_{o} \in {X}... | Yes |
Proposition 14.1 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space. A set \( E \subset X \) is weakly bounded if and only if is strongly bounded. | Proof If \( E \) is strongly bounded, there exists a constant \( R \) such that \( \parallel x\parallel \leq R \) for all \( x \in E \) . Then for all \( T \in {X}^{ * } \n\n\[ \n\left| {T\left( x\right) }\right| \leq \parallel T\parallel \parallel x\parallel \leq \parallel T\parallel R. \n\] \n\nThus \( E \) is weakly... | Yes |
Corollary 14.1 Let \( E \) be a weakly, countably compact subset of a normed space \( \{ X;\parallel \cdot \parallel \} \) . Then \( E \) is strongly bounded. | Proof Let \( \mathcal{O} \) be a weakly open neighborhood of the origin. Then the collection \( \{ n\mathcal{O}{\} }_{n \in \mathbb{N}} \) is a countable, weakly open covering for \( E \), since \( X = \bigcup n\mathcal{O} \) . Therefore \( E \subset t\mathcal{O} \) for some \( t > 0 \) . Thus \( E \) is weakly bounded... | Yes |
Corollary 14.2 Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space and let \( \left\{ {x}_{n}\right\} \) be a sequence of elements of \( X \) weakly convergent to some \( x \in X \). There exists a positive constant \( C \) such that \( \begin{Vmatrix}{x}_{n}\end{Vmatrix} \leq C \) for all \( n \). Moreover\n... | Proof The uniform upper bound of \( \begin{Vmatrix}{x}_{n}\end{Vmatrix} \) follows from Proposition 14.1. For all \( T \in {X}^{ * } \), by definition of weak limit\n\n\[ \left| {T\left( x\right) }\right| \leq \liminf \left| {T\left( {x}_{n}\right) }\right| \leq \parallel T\parallel \liminf \begin{Vmatrix}{x}_{n}\end{V... | Yes |
Proposition 14.2 Let \( \{ X;\parallel \cdot \parallel \} \) be a reflexive Banach space. Then every bounded sequence \( \left\{ {x}_{n}\right\} \) of elements of \( X \) contains a weakly convergent subsequence \( \left\{ {x}_{{n}^{\prime }}\right\} \) . | Proof Let \( {X}_{o} \) be the closed linear span of \( \left\{ {x}_{n}\right\} \) . Such a subspace is separable since the finite linear combinations of elements of \( \left\{ {x}_{n}\right\} \) with rational coefficients is a countable dense subset.\n\nSince \( {X}_{o} \) is reflexive, its double-dual \( {X}_{o}^{* *... | Yes |
Corollary 14.3 Let \( \{ X;\parallel \cdot \parallel \} \) be a reflexive Banach space. A subset \( C \subset X \) is weakly sequentially compact if and only if is both weakly bounded and weakly sequentially closed. | Proof Sequential compactness implies countable compactness (Proposition 5.2 of Chap. 2). | No |
Theorem 16.1 (Alaoglu [2]) Let \( \{ X;\parallel \cdot \parallel \} \) be a normed space and let \( {X}^{ * } \) be its dual. The closed unit ball \( {B}^{ * } = \left\{ {T \in {X}^{ * } \mid \parallel T\parallel \leq 1}\right\} \) in \( {X}^{ * } \), is weak* compact. | Proof If \( T \in {B}^{ * } \) then \( T\left( x\right) \in \left\lbrack {-\parallel x\parallel ,\parallel x\parallel }\right\rbrack \) for all \( x \in X \) . Consider now the Cartesian product\n\n\[ P = \mathop{\prod }\limits_{{x \in X}}\left\lbrack {-\parallel x\parallel ,\parallel x\parallel }\right\rbrack \]\n\nA ... | Yes |
Lemma 16.1 These two topologies coincide on \( {B}^{ * } \) . | Proof Every weak* open neighborhood of a point \( {T}_{o} \in {X}^{ * } \) contains an open set of the form\n\n\[ \mathcal{O} = \left\{ \begin{matrix} T \in {X}^{ * }\left| \right| T\left( {x}_{j}\right) - {T}_{o}\left( {x}_{j}\right) \mid < \delta \text{ for some }\delta > 0 \\ \text{ for finitely many }{x}_{j}, j = 1... | Yes |
Lemma 16.2 \( {B}^{ * } \) is closed in its relative product topology. | Proof Let \( {f}_{o} \) be in the closure of \( {B}^{ * } \) in the relative product topology. Fix \( x, y \in X \) and \( \alpha ,\beta \in \mathbb{R} \) and consider the three points\n\n\[ \n{x}_{1} = x\;{x}_{2} = y\;{x}_{3} = {\alpha x} + {\beta y}.\n\]\n\nFor \( \varepsilon > 0 \), the sets\n\n\[ \n{\mathcal{V}}_{\... | Yes |
Proposition 18.1 (Riesz [133]) Let \( {H}_{o} \) be a closed, convex, proper subset of \( H \) . Then for every \( x \in H - {H}_{o} \) there exists a unique \( {x}_{o} \in {H}_{o} \), such that\n\n\[ \mathop{\inf }\limits_{{y \in {H}_{o}}}\parallel x - y\parallel = \begin{Vmatrix}{{x}_{o} - x}\end{Vmatrix}. \]\n\n(18.... | Proof Let \( \left\{ {y}_{n}\right\} \) be a sequence in \( {H}_{o} \) such that\n\n\[ \delta \overset{\text{ def }}{ = }\mathop{\inf }\limits_{{y \in {H}_{o}}}\parallel x - y\parallel = \lim \begin{Vmatrix}{{y}_{n} - x}\end{Vmatrix}. \]\n\n(18.2)\n\nSince \( {H}_{o} \) is convex\n\n\[ \frac{{y}_{n} + {y}_{m}}{2} \in {... | Yes |
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