Questions and Answers of Measures Integrals And Martingales

Find \(\mathbb{1}_{[0,1]} * \mathbb{1}_{[0,1]}\) and \(\mathbb{1}_{[0,1]} * \mathbb{1}_{[0,1]} * \mathbb{1}_{[0,1]}\).
Show that \(\operatorname{supp}(u * w) \subset \overline{\operatorname{supp} u+\operatorname{supp} w},(A+B:=\{a+b: a \in A, b \in B\})\).
(Mellin convolution in the group \(((0, \infty), \cdot)\) ) Define on \(((0, \infty), \mathscr{B}(0, \infty))\) the measure \(\mu(d x)=x^{-1} d x\). The Mellin convolution of measurable \(u, w:(0,
Let \((X, \mathscr{A}, \mu)\) be a measure space and \((Y, \mathscr{B})\) be a measurable space. Assume that \(T: A ightarrow B, A \in \mathscr{A}, B \in \mathscr{B}\), is an invertible measurable
Let \(\mu\) be a measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)\) and \(x, y, z \in \mathbb{R}^{n}\). Find \(\delta_{x} \star \delta_{y}\) and \(\delta_{z} \star \mu\).
Let \(\mu, u\) be two \(\sigma\)-finite measures on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight.\) ). Show that \(\mu \star u\) has no atoms (see Problem 6.8) if \(\mu\) has no
Let \(\mathbb{P}\) be a probability measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)\) and denote by \(\mathbb{P}^{\star n}\) the \(n\)-fold convolution product
Let \(p: \mathbb{R} ightarrow \mathbb{R}\) be a polynomial and \(u \in C_{c}(\mathbb{R})\). Show that \(u \star p\) exists and is again a polynomial.
Let \(w: \mathbb{R} ightarrow \mathbb{R}\) be an increasing (and hence measurable, by Problem 8.21 ) and bounded function. Show that for every positive \(u \in \mathcal{L}^{1}\left(\lambda^{1}ight)\)
Assume that \(u \in C_{c}\left(\mathbb{R}^{n}ight)\) and \(w \in C^{\infty}\left(\mathbb{R}^{n}ight)\). Show that \(u \star w\) exists, is of class \(C^{\infty}\) and
Modify the proof of Theorem 15.11 and show that \(C_{c}^{\infty}\left(\mathbb{R}^{n}ight)\) is uniformly dense in \(C_{c}\left(\mathbb{R}^{n}ight)\).Data from theorem 15.11 Theorem 15.11 Let X" be
Young's inequality. Adapt the proof of Theorem 15.6 and show that\[\|u \star w\|_{r} \leqslant\|u\|_{p} \cdot\|w\|_{q}\]for all \(p, q, r \in[1, \infty), u \in \mathcal{L}^{p}\left(\lambda^{n}ight),
A general Young inequality. Generalize Young's inequality given in Problem 15.14 and show that\[\left\|f_{1} \star f_{2} \star \cdots \star f_{N}ight\|_{r} \leqslant
Define \(\phi: \mathbb{R} ightarrow \mathbb{R}\) by \(\phi(x):=(1-\cos x) \mathbb{1}_{[0,2 \pi)}(x)\), let \(u(x):=1, v(x):=\phi^{\prime}(x)\) and \(w(x):=\int_{(-\infty, x)} \phi(t) d t\). Then(i)
Let \(F, F_{1}, F_{2}, F_{3}, \ldots\) be \(F_{\sigma}\)-sets in \(\mathbb{R}^{n}\). Show that(i) \(F_{1} \cap F_{2} \cap \cdots \cap F_{N}\) is for every \(N \in \mathbb{N}\) an
Prove the following corollary to Lemma 16.12 : Lebesgue measure \(\lambda^{n}\) on \(\mathbb{R}^{n}\) is outer regular, i.e.\[\lambda^{n}(B)=\inf \left\{\lambda^{n}(U): U \supset B, U \text { open
Completion (6). Combine Problems 16.2 and 11.6 to show that the completion \(\bar{\lambda}^{n}\) of \(n\)-dimensional Lebesgue measure is again inner and outer regular.Data from problem 16.2Prove the
Let \(C\) be Cantor's ternary set, see page 4 and Problem 7.12.(i) Show that \(C-C:=\{x-y: x, y \in C\}\) is the interval \([-1,1]\).(ii) Show that this proves that the result of Corollary 16.14 is
Consider the Borel \(\sigma\)-algebra \(\mathscr{B}[0, \infty)\) and write \(\lambda=\left.\lambda^{1}ight|_{[0, \infty)}\) for Lebesgue measure on the half-line \([0, \infty)\).(i) Show that
Use Jacobi's transformation formula to recover Theorem 5.8(i), Problem 5.9 and Theorem 7.10. In particular, for all integrable functions \(u: \mathbb{R}^{n} ightarrow[0,
Arc-length. Let \(f: \mathbb{R} ightarrow \mathbb{R}\) be a twice continuously differentiable function and denote by \(\Gamma_{f}:=\{(t, f(t)): t \in \mathbb{R}\}\) its graph. Define a function
Let \(\Phi: \mathbb{R}^{d} ightarrow M \subset \mathbb{R}^{n}, d \leqslant n\), be a \(C^{1}\)-diffeomorphism.(i) Show that \(\lambda_{M}:=\Phi\left(|\operatorname{det} D \Phi| \lambda^{d}ight)\) is
In Example 12.15 we introduced Euler's gamma function:\[\Gamma(t)=\int_{(0, \infty)} x^{t-1} e^{-x} \lambda^{1}(d x)\]Show that \(\Gamma\left(\frac{1}{2}ight)=\sqrt{\pi}\).Data from example 12.15
3D polar coordinates. Define \(\Phi:[0, \infty) \times[0,2 \pi) \times[-\pi / 2, \pi / 2) ightarrow \mathbb{R}^{3}\) by\[\Phi(r, \theta, \omega):=(r \cos \theta \cos \omega, r \sin \theta \cos
Euler's Inegrals. Euler's gamma and beta functions are the following parameterdependent integrals for \(x, y>0\) :\[\Gamma(x):=\int_{0}^{\infty} e^{-t} t^{x-1} d t \quad \text { and } \quad B(x,
Compute for \(m, n \in \mathbb{N}\) the integral\[\int_{B_{1}(0)} x^{m} y^{n} d(x, y)\]
Let \((X, \mathscr{A}, \mu)\) be a measure space. Assume that \(\mathcal{D} \subset \mathcal{L}^{p}(\mu)\) is dense. If \(\mathcal{C} \subset \mathcal{D}\) is dense w.r.t. the norm
The following exercise provides an independent proof of Theorem 17.12 in \(\mathbb{R}^{n}\). Set \(d(x, A):=\inf _{a \in A}|x-a|\) and assume that all measures are finite on compact sets.(i)
Consider on \((\mathbb{R}, \mathscr{B}(\mathbb{R}), d x)\) the Lebesgue space \(\mathcal{L}^{p}(d x), \quad 1 \leqslant p
Denote by \(d x\) one-dimensional Lebesgue measure and let \(f \in \mathcal{L}^{1}(d x)\). Define the mean value\[M_{h} f(x):=\frac{1}{2 h} \int_{x-h}^{x+h} f(t) d t\]and show that(i) \(M_{h}
Let \(\mu\) be an outer regular measure on \((X, d, \mathscr{B}(X)), 1 \leqslant p(i) Let \(A \in \mathscr{B}(X)\) such that \(f=\mathbb{1}_{A} \in \mathcal{L}^{p}(\mu)\). Show that for every
Let \((X, d)\) be a metric space which is separable (i.e. it contains a countable dense subset) and locally compact (i.e. every \(x \in X\) has an open neighbourhood \(U\) such that \(\bar{U}\) is
Lusin's theorem. The following steps furnish a proof of the following result.Theorem (Lusin). Let \(\mu\) be an outer regular measure on the space \((X, d, \mathscr{B}(X))\). For every \(f \in
Consider Lebesgue measure \(\lambda\) on the space \(([a, b], \mathscr{B}[a, b])\) and assume there on that \(f \in \mathcal{L}^{1}([a, b], \lambda)\) satisfies \(\int_{a}^{b} x^{n} f(x) d x=0\)
Show that the outer regularity from Corollary 18.10 coincides with the usual notion, i.e.\[\overline{\mathcal{H}}^{\phi}(A)=\inf \left\{\mathcal{H}^{\phi}(U): U \supset A, U \text { open
Finish the proof of Theorem 18.12 (i) and show that every set \(A \subset \mathbb{R}^{n}\) is indeed \(\overline{\mathcal{H}}^{0}\)-measurable.Data from theorem 18.12 Theorem 18.12 Let Hs, s20 be
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and let \(u\) be a further measure. Show that \(u \leqslant \mu\) entails that \(u=f \mu\) for some (a.e. uniquely determined)
Let \(\mu, u\) be two \(\sigma\)-finite measures on \((X, \mathscr{A})\) which have the same null sets. Show that \(u=f \mu\) and \(\mu=g u\), where \(0
Give an example of a measure \(\mu\) and a density \(f\) such that \(f \mu\) is not \(\sigma\)-finite.
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and assume that \(u=f \mu\) for a positive measurable function \(f\).(i) Show that \(u\) is a finite measure if, and only if, \(f
Two measures \(ho, \sigma\) defined on the same measurable space \((X, \mathscr{A})\) are singular, if there is a set \(N \in \mathscr{A}\) such that \(ho(N)=\sigma\left(N^{c}ight)=0\). If this is
Bounded variation and absolute continuity. Let \(\lambda\) be one-dimensional Lebesgue measure.A function \(F:[a, b] ightarrow \mathbb{R}\) is said to beabsolutely continuous (AC) if for each
Let \(\mu\) be Lebesgue measure on \([0,2]\) and \(u\) be Lebesgue measure on \([1,3]\). Find the Lebesgue decomposition of \(u\) with respect to \(\mu\).
Stielties measure (3). Let \((\mathbb{R}, \mathscr{B}(\mathbb{R}), \mu)\) be a finite measure space and denote by \(F\) the left-continuous distribution function of \(\mu\) as in Problem 6.1. Use
The devil's staircase. Recall the construction of Cantor's ternary set from Problem 7.12. Denote by \(I_{n}^{1}, \ldots, I_{n}^{n}-1\) the intervals which make up \([0,1] \backslash C_{n}\) arranged
The indicator function of a set \(A \subset X\) is defined by\[\mathbb{1}_{A}(x):= \begin{cases}1 & \text { if } x \in A \\ 0 & \text { if } x otin A\end{cases}\]Check that for \(A, B, A_{i} \subset
Let \(A, B, C \subset X\) and denote by \(A \triangle B\) the symmetric difference as in Problem 2.2 . Show that(i)(ii) \(A \Delta(B \triangle C)=(A \triangle B) \triangle C\);(iii)
Let \(f: X ightarrow Y\) be a map, \(A \subset X\) and \(B \subset Y\). Show that, in general,\[f \circ f^{-1}(B) \varsubsetneqq B \text { and } f^{-1} \circ f(A) \supsetneq A \text {. }\]When does '
Show that \(\{0,1\}^{\mathbb{N}}=\{\) all infinite sequences consisting of 0 and 1\(\}\) is uncountable.
Show that the set \(\mathbb{R}\) is uncountable and that \(\#(0,1)=\# \mathbb{R}\).[find a bijection \(f:(0,1) ightarrow \mathbb{R}\).]
Adapt the proof of Theorem 2.8 to show that \(\#\{1,2\}^{\mathbb{N}} \leqslant \#(0,1) \leqslant \#\{0,1\}^{\mathbb{N}}\) and conclude that \(\#(0,1)=\#\{0,1\}^{\mathbb{N}}\).Remark. This is the
Extend Problem 2.15 to deduce \(\#\{0,1,2, \ldots, n\}^{\mathbb{N}}=\#(0,1)\) for all \(n \in \mathbb{N}\).Data from problem 2.15Adapt the proof of Theorem 2.8 to show that \(\#\{1,2\}^{\mathbb{N}}
Mimic the proof of Theorem 2.9 to show that \(\#(0,1)^{2}=\mathfrak{c}\). Use the fact that \(\# \mathbb{R}=\#(0,1)\) to conclude that \(\# \mathbb{R}^{2}=\mathfrak{c}\).Data from theorem 2.9 We have
Show that the set of all infinite sequences of natural numbers \(\mathbb{N}^{\mathbb{N}}\) has cardinality \(\mathfrak{c}\). [use that
Let \(A_{1}, A_{2}, \ldots, A_{N}\) be non-empty subsets of \(X\).(i) If the \(A_{n}\) are disjoint and \(\biguplus A_{n}=X\), then \(\# \sigma\left(A_{1}, A_{2}, \ldots, A_{N}ight)=2^{N}\).Remark. A
Find an example (e.g. in \(\mathbb{R}\) ) showing that \(\bigcap_{n \in \mathbb{N}} U_{n}\) need not be open even if all \(U_{n}\) are open sets.
Prove any one of the assertions made in Remark 3.9 .Data from remark 3.9 Let D be a dense subset of R, e.g. D=Q or D=R. The Borel sets of the real line R are also generated by any of the following
Check that the set functions defined in Example 4.5 are measures in the sense of Definition 4.1 .Data From example 4.5Data from definition 4.1 (i) (Dirac measure, unit mass) Let (X, A) be a
.Let \((X, \mathscr{A})\) be a measurable space.(i). Let \(\mu, u\) be two measures on \((X, \mathscr{A})\). Show that the set function \(ho(A):=a \mu(A)+\) \(b u(A), A \in \mathscr{A}\), for all
. Let \((X, \mathscr{A}, \mu)\) be a finite measure space and \(\left(A_{n}ight)_{n \in \mathbb{N}},\left(B_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) such that \(A_{n} \supset B_{n}\) for all
Let \((X, \mathscr{A}, \mu)\) be a measure space, let \(\mathscr{F} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra and denote the collection of all \(\mu\)-null sets by \(\mathscr{N}=\{N \in
Let \(\overline{\mathscr{A}}\) denote the completion of \(\mathscr{A}\) as in Problem 4.15 and write\[\mathscr{N}:=\{N \subset X: \exists M \in \mathscr{A}, N \subset M, \mu(M)=0\}\]for the family of
Regularity on Polish spaces. A Polish space XX is a complete metric space which has a countable dense subset D⊂XD⊂X.Let μμ be a finite measure on (X,B(X))(X,B(X)). Then μμ is regular in the
An alternative definition of Dynkin systems. A family \(\mathscr{F} \subset \mathscr{P}(X)\) is a Dynkin system if, and only if,Conclude that any Dynkin system is a monotone class in the sense of
Monotone classes (2). Recall from Problem 3.14 that a monotone class \(\mathscr{M} \subset \mathscr{P}(X)\) is a family which contains \(X\) and is stable under countable unions of increasing sets
(i) Show that non-void open sets in \(\mathbb{R}\) (resp. \(\mathbb{R}^{n}\) ) have always strictly positive Lebesgue measure.[let \(U\) be open. Find a small ball in \(U\) and inscribe a
(i) Show that \(\lambda^{1}((a, b))=b-a\) for all \(a, b \in \mathbb{R}, a \leqslant b\).[ approximate \((a, b)\) by half-open intervals and use Proposition 4.3 (vi), (vii).](ii) Let \(H \subset
Let \(\lambda:=\left.\lambda^{1}ight|_{[0,1]}\) be Lebesgue measure on \(([0,1], \mathscr{B}[0,1])\). Show that for every \(\epsilon>0\) there is a dense open set \(U \subset[0,1]\) with \(\lambda(U)
Let \(\lambda=\lambda^{1}\) be Lebesgue measure on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\). Show that \(N \in \mathscr{B}(\mathbb{R})\) is a null set if, and only if, for every \(\epsilon>0\)
Borel-Cantelli lemma (1) - the direct half. Prove the following theorem.Theorem (Borel-Cantelli lemma). Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. For every sequence
Non-measurable sets (1). Let \(\mu\) be a measure on \(\mathscr{A}=\{\emptyset,[0,1),[1,2),[0,2)\}, X=[0,2)\), such that \(\mu([0,1))=\mu([1,2))=\frac{1}{2}\) and \(\mu([0,2))=1\). Denote by
Non-measurable sets (2). Consider on \(X=\mathbb{R}\) the \(\sigma\)-algebra \(\mathscr{A}:=\left\{A \subset \mathbb{R}: Aight.\) or \(A^{c}\) is countable \} from Example 3.3 (v) and the measure
Use Lemma 7.2 to show that \(\tau_{x}: y \mapsto y-x, x, y \in \mathbb{R}^{n}\), is \(\mathscr{B}\left(\mathbb{R}^{n}ight) / \mathscr{B}\left(\mathbb{R}^{n}ight)\)-measurable.Data from lemma 7.2 Let
Show that \(\Sigma^{\prime}\) defined in the proof of Lemma 7.2 is a \(\sigma\)-algebra.Data from lemma 7.2 Let (X, A), (X', A') be measurable spaces and let A' = o(9'). Then T:XX' is A/A'-measurable
Let \(X=\mathbb{Z}=\{0, \pm 1, \pm 2, \ldots\}\). Show that(i) \(\mathscr{A}:=\{A \subset \mathbb{Z} \mid \forall n>0: 2 n \in A \Longleftrightarrow 2 n+1 \in A\}\) is a \(\sigma\)-algebra;(ii) \(T:
Let \(X\) be a set, let \(\left(X_{i}, \mathscr{A}_{i}ight), i \in I\), be arbitrarily many measurable spaces and let \(T_{i}: X ightarrow X_{i}\) be a family of maps.(i) Show that for every \(i \in
Let \((X, \mathscr{A})\) and \(\left(X^{\prime}, \mathscr{A}^{\prime}ight)\) be measurable spaces and \(T: X ightarrow X^{\prime}\).(i) Show that
Let \(T: X ightarrow Y\) be any map. Show that \(T^{-1}(\sigma(\mathscr{G}))=\sigma\left(T^{-1}(\mathscr{G})ight)\) holds for arbitrary families \(\mathscr{G}\) of subsets of \(Y\).
Let \(X\) be a set, let \(\left(X_{i}, \mathscr{A}_{i}ight), i \in I\), be arbitrarily many measurable spaces and let \(T_{i}: X ightarrow X_{i}\) be a family of maps. Show that a map \(f\) from a
Use Problem 7.7 to show that a function \(f: \mathbb{R}^{n} ightarrow \mathbb{R}^{m}, x \mapsto\left(f_{1}(x), \ldots, f_{m}(x)ight)\) is measurable if, and only if, all coordinate maps \(f_{i}:
Let \(T:(X, \mathscr{A}) ightarrow\left(X^{\prime}, \mathscr{A}^{\prime}ight)\) be a measurable map. Under which circumstances is the family of sets \(T(\mathscr{A})\) a \(\sigma\)-algebra?
Use image measures to give a new proof of Problem 5.9 , i.e. show that\[\lambda^{n}(t \cdot B)=t^{n} \lambda^{n}(B) \quad \forall B \in \mathscr{B}\left(\mathbb{R}^{n}ight), \quad \forall t>0\]Data
Let \((X, \mathscr{A})=(\mathbb{R}, \mathscr{B}(\mathbb{R}))\) and let \(\lambda\) be one-dimensional Lebesgue measure.(i) A point \(x\) with \(\mu\{x\}>0\) is an atom. Show that every measure
Cantor's ternary set. Let \((X, \mathscr{A})=([0,1],[0,1] \cap \mathscr{B}(\mathbb{R})), \lambda=\left.\lambda^{1}ight|_{[0,1]}\), and set \(C_{0}=[0,1]\). Remove the open middle third of \(C_{0}\)
Let \(\mathcal{E}, \mathscr{F} \subset \mathscr{P}(X)\) be two families of subsets of \(X\). One usually uses the notation (as we do in this book)\[\mathcal{E} \cup \mathscr{F}=\{A: A \in \mathcal{E}
Show directly that condition (i) of Lemma 8.1 is equivalent to one of the conditions (ii), (iii) and (iv).Data from lemma 8.1 Let (X,A) be a measurable space. The function u: XR is A/B(R)-measurable
Verify that \(\mathscr{B}(\overline{\mathbb{R}})\) defined in (8.5) is a \(\sigma\)-algebra. Show that \(\mathscr{B}(\mathbb{R})=\mathbb{R} \cap \mathscr{B}(\overline{\mathbb{R}})\).Equation 8.5 B* E
Let \((X, \mathscr{A})\) be a measurable space.(i) Let \(f, g: X ightarrow \mathbb{R}\) be measurable functions. Show that for every \(A \in \mathscr{A}\) the function \(h(x):=f(x)\), if \(x \in A\),
Let \((X, \mathscr{A})\) be a measurable space and let \(\mathscr{B} \varsubsetneqq \mathscr{A}\) be a sub- \(\sigma\)-algebra. Show that \(\mathcal{M}(\mathscr{B}) \varsubsetneqq
Show that \(f \in \mathcal{E}\) implies that \(f^{ \pm} \in \mathcal{E}\). Is the converse true?
Show that for every real-valued function \(u=u^{+}-u^{-}\)and \(|u|=u^{+}+u^{-}\).
Show that \(x \mapsto \max \{x, 0\}\) and \(x \mapsto \min \{x, 0\}\) are continuous, and by virtue of Problem 8.7 or Example 7.3 , measurable functions from \(\mathbb{R} ightarrow \mathbb{R}\).
Let \(\left(f_{i}ight)_{i \in I}\) be arbitrarily many maps \(f_{i}: X ightarrow \mathbb{R}\). Show that(i) \(\left\{\sup _{i} f_{i}>\lambdaight\}=\bigcup_{i}\left\{f_{i}>\lambdaight\}\);(iii)
Check that the approximating sequence \(\left(f_{n}ight)_{n \in \mathbb{N}}\) for \(u\) in Theorem 8.8 consists of \(\sigma(u)\)-measurable functions.Data from theorem 8.8 (sombrero lemma) Let (X, A)
Complete the proofs of Corollaries 8.12 and 8.13.Data from corollaries 8.12Data from corollaries 8.13 A function u is A/B(R)-measurable if, and only if, the positive and negative parts ut are
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and assume that \(\mathscr{A}\) is generated by a Boolean algebra \(\mathscr{G}\), i.e. a family of subsets of \(X\) which is stable
Let \(u: \mathbb{R} ightarrow \mathbb{R}\) be differentiable. Explain why \(u\) and \(u^{\prime}=d u / d x\) are measurable.
Find \(\sigma(u)\), i.e. the \(\sigma\)-algebra generated by \(u\), for the following functions:\(f, g, h: \mathbb{R} ightarrow \mathbb{R}\),(i) \(f(x)=x\);(ii) \(g(x)=x^{2} ;\)(iii)
Let \(\lambda\) be one-dimensional Lebesgue measure. Find \(\lambda \circ u^{-1}\), if \(u(x)=|x|\).
Let \(E \in \mathscr{B}(\mathbb{R}), Q: E ightarrow \mathbb{R}, Q(x)=x^{2}\), and \(\lambda_{E}:=\lambda(E \cap \cdot)\) (Lebesgue measure).(i) Show that \(Q\) is \(\mathscr{B}(E) /

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