Measures Integrals And Martingales 2nd Edition René L. Schilling - Solutions

For every \(B \subset \mathbb{R}^{n}\) one has \(\operatorname{dim}_{\mathcal{H}} B \leqslant n\). If \(B\) contains an open set or a set of strictly positive Lebesgue measure, then \(\operatorname{dim}_{\mathcal{H}} B=n\).
Consider the fractals introduced in Problems 1.4 and 1.5: the Koch snowflake and the Sierpiński triangle. Give a heuristic argument that the Hausdorff dimension of the Sierpiński triangle is \(\log 3 / \log 2\). What is the dimension of the Koch snowflake?Data from problem 1.4 Let \(K_{0}
Assume that \(\phi, \psi:[0, \infty) ightarrow[0, \infty)\) are admissible for the construction of Hausdorff measures and assume that \(\lim _{x ightarrow 0} \phi(x) / \psi(x)=0\). Show that \(\overline{\mathcal{H}}^{\psi}(A)
Calculate the Fourier transform of the following functions/measures on \(\mathbb{R}\) :(a) \(\mathbb{1}_{[-1,1]}(x)\),(b) \(\mathbb{1}_{[-1,1]} \star \mathbb{1}_{[-1,1]}(x)\),(c) \(e^{-x} \mathbb{1}_{[0, \infty)}(x)\),(d) \(e^{-|x|}\),(e) \(1 /\left(1+x^{2}ight)\),(f) \(\quad(1-|x|)
Extend Plancherel's theorem (Theorem 19.20 ) to show that\[\int \widehat{u}(\xi) \overline{\widehat{v}(\xi)} d \xi=(2 \pi)^{-n} \int u(x) \overline{v(x)} d x \quad \forall u, v \in L_{\mathbb{C}}^{2}\left(\lambda^{n}ight)\]Data from theorem 19.20 Theorem 19.20 (Plancherel) If u = L(X") L(X"), then
Let \(\mu\) be a finite measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)\) and set \(\chi(\xi):=\widehat{\mu}(\xi)\). Show that \(\chi\) is real-valued if, and only if, \(\mu\) is symmetric w.r.t. the origin, i.e. \(\widetilde{\mu}(B):=\mu(-B)=\mu(B)\).
Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric, positive definite matrix. Find the Fourier transform of the function \(e^{-\langle x, A xangle}\).
Assume that \(u \in L^{1}\left(\lambda^{1}ight) \cap L^{\infty}\left(\lambda^{1}ight)\) and \(\widehat{u} \geqslant 0\). Show that \(\widehat{u} \in L^{1}\left(\lambda^{1}ight)\). Extend the assertion to \(u \in L^{2}\left(\lambda^{1}ight)\)[estimate \(\int \widehat{u} \widehat{g}_{t} d \xi\),
Let \(\mu\) be a finite measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)\). Prove P. Lévy's truncation inequality:\[\mu\left(\mathbb{R}^{n} \backslash[-2 R, 2 R]^{n}ight) \leqslant 2\left(\frac{R}{2}ight)^{n} \int_{[-1 / R, 1 /
Let \(\mu\) be a finite measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)\) and denote by \(\phi(\xi):=\widehat{\mu}(\xi)\) the Fourier transform(i) \(\phi\) is positive semidefinite, i.e. for all \(m \in \mathbb{N}, \xi_{1}, \ldots, \xi_{m} \in \mathbb{R}^{n},
Let \(B \in \mathscr{B}(\mathbb{R})\). If \(\int_{B} e^{i x / n} d x=0\) for all \(n=1,2, \ldots\), then \(\lambda^{1}(B)=0\).
Assume that \(\mu\) is a finite measure on \((\mathbb{R}, \mathscr{B}(\mathbb{R})\) ). Show that(i) \(\exists \xi eq 0: \widehat{\mu}(\xi)=\widehat{\mu}(0) \Longleftrightarrow \exists \xi eq 0: \mu(\mathbb{R} \backslash(2 \pi / \xi) \mathbb{Z})=0\);(ii) \(\exists \xi_{1}, \xi_{2}, \xi_{1} / \xi_{2}
Where was \(\sigma\)-additivity used when calculating the length of the Cantor set?
Let \(K_{0} \subset \mathbb{R}^{2}\) be a line of length 1 . We get \(K_{1}\) by replacing the middle third of \(K_{0}\) by the sides of an equilateral triangle. By iterating this procedure we get the curves \(K_{0}, K_{1}, K_{2}, \ldots\) (see Fig. 1.5 ) which defines in the limit Koch's snowflake
Let \(S_{0} \subset \mathbb{R}^{2}\) be a solid equilateral triangle. We get \(S_{1}\) by removing the middle triangle whose vertices are the midpoints of the sides of \(S_{1}\). By repeating this procedure with the four triangles which make up \(S_{1}\) etc. we get \(S_{0}, S_{2}, S_{2}, \ldots\)
Let \(A, B, C \subset X\) be sets. Show that(i) \(A \backslash B=A \cap B^{c}\);(ii) \((A \backslash B) \backslash C=A \backslash(B \cup C)\);(iii) \(A \backslash(B \backslash C)=(A \backslash B) \cup(A \cap C)\);(iv) \(A \backslash(B \cap C)=(A \backslash B) \cup(A \backslash C)\);(v) \(A
Prove de Morgan's identities (2.2) and (2.3).Equation 2.2Equation 2.3 (An B)=A U B, (AUB)=An B,
(i). Find examples showing that \(f(A \cap B) eq f(A) \cap f(B)\) and \(f(A \backslash B) eq f(A) \backslash f(B)\). In both relations one inclusion ' \(C\) ' or ' \(\supset\) ' is always true. Which one? (ii).Prove (2.6).Equation 2.6 S(UG)=US(c), El iEl f'(na)=n(a), EI El f'(C\D)=f(C) \ (D). $S(UG)=US(c), El iEl f'(na)=n(a), EI El f'(C\D)=f(C) \ (D).$
Let \(f\) and \(g\) be two injective maps. Show that \(f \circ g\), if it exists, is injective.
Consider on \(\left(\mathbb{R}^{n}, \mathcal{B}\left(\mathbb{R}^{n}ight)ight.\) ) the Dirac measure \(\delta_{x}\) for some fixed \(x \in \mathbb{R}^{n}\). Find the completion of \(\mathcal{B}\left(\mathbb{R}^{n}ight)\) with respect to \(\delta_{x}\).
Restriction. Let \((X, \mathscr{A}, \mu)\) be a measure space and let \(\mathscr{B} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra. Denote by \(u:=\left.\muight|_{\mathscr{B}}\) the restriction of \(\mu\) to \(\mathscr{B}\).(i) Show that \(u\) is again a measure.(ii) Assume that \(\mu\) is a
Show that a measure space \((X, \mathscr{A}, \mu)\) is \(\sigma\)-finite if, and only if, there exists a sequence of measurable sets \(\left(E_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) such that \(\bigcup_{n \in \mathbb{N}} E_{n}=X\) and \(\mu\left(E_{n}ight)
Regularity. Let XX be a metric space and μμ be a finite measure on the Borel sets B=B(X)B=B(X) and denote the open sets by OO and the closed sets by FF. Define a family of

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