Questions and Answers of Measures Integrals And Martingales

Fatou's lemma for measures. Let \((X, \mathscr{A}, \mu)\) be a measure space and let \(\left(A_{n}ight)_{n \in \mathbb{N}}, A_{n} \in \mathscr{A}\), be a sequence of measurable sets. We set
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(u \in \mathcal{L}^{1}(\mu)\). Show that for every \(\epsilon>0\) there is some \(\delta>0\) such that\[A \in \mathscr{A}, \mu(A)
Let\[f(t)=\int_{0}^{\infty} \arctan \left(\frac{t}{\sinh x}ight) d x, \quad t>0\]where \(\sinh x=\frac{1}{2}\left(e^{x}-e^{-x}ight)\).(i) Show that \(f\) is differentiable on \((0, \infty)\), but
Let \(\left(A_{i}ight)_{i \in \mathbb{N}}\) be a sequence of sets of cardinality (\mathfrak{c}\). Show that \(\# \bigcup_{i \in \mathbb{N}} A_{i}=\mathfrak{c}\).[map \(A_{i}\) bijectively onto
Alternative characterization of \(\mathscr{B}\left(\mathbb{R}^{n}ight)\). In older books the Borel sets are often introduced as the smallest family \(\mathscr{M}\) of sets which is stable under
Completion (1) We have seen in Problem 4.12 that measurable subsets of null sets are again null sets: \(M \in \mathscr{A}, M \subset N \in \mathscr{A}, \mu(N)=0\) then \(\mu(M)=0\); but there might
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\mathscr{H} \subset \mathscr{G}\) be two sub- \(\sigma\)-algebras. Show that\[\mathbb{E}^{\mathscr{H}} \mathbb{E}^{\mathscr{G}}
Let \(u \in \mathcal{L}^{2}(\mu)\) be a positive, integrable function. Show that \(\int_{\{u>1\}} u d \mu \leqslant \mu\{u>1\}\) entails that \(u(x) \leqslant 1\) for \(\mu\)-a.e. \(x\).Remark.
Let \((X, \mathscr{A}, \mu)\) be a measure space, \(\mathscr{G} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra and let \(u:=f \mu\) where \(f \in \mathcal{M}^{+}(\mathscr{A})\) is a density
Let \((X, \mathscr{A}, \mu)\) be a finite measure space, \(G_{1}, \ldots, G_{n} \in \mathscr{A}\) such that \(\bigcup_{i=1}^{n} G_{i}=X\) and \(\mu\left(G_{i}ight)>0\) for all \(i=1,2, \ldots, n\).
Extension by continuity. Let \(T: L^{2}(\mu) ightarrow L^{2}(\mu)\) be a linear operator such that \(\|T u\|_{p} \leqslant\) \(c\|u\|_{p}\) for \(u \in L^{2}(\mu) \cap L^{p}(\mu)\) for some \(p eq
Let \((X, \mathscr{A}, \mu)\) be a measure space and let \(\mathscr{G} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra such that \(\left.\muight|_{\mathscr{G}}\) is \(\sigma\)-finite. Let \(p, q
Complete the proof of Theorem 27.11.Data from theorem 27.11 Theorem 27.11 Let (X, A,p) be a o-finite measure space and let HCGCA be sub-o-algebras. The conditional expectation E has the following
Show that \(\mathbb{E}^{\mathscr{G}} 1=1\) if, and only if, \(\left.\muight|_{\mathscr{G}}\) is \(\sigma\)-finite. Find a counterexample showing that \(\mathbb{E}^{\mathscr{G}} 1 \leqslant 1\) is, in
Let \(\mathscr{G}\) be a sub- \(\sigma\)-algebra of \(\mathscr{A}\). Show that \(\mathbb{E}^{\mathscr{G}} g=g\) for all \(g \in L^{p}(\mathscr{G})\).[observe that, a.e., \(g=g
Let \(\mathscr{H} \subset \mathscr{G}\) be two sub- \(\sigma\)-algebras of \(\mathscr{A}\). Show that\[\mathbb{E}^{\mathscr{G}} \mathbb{E}^{\mathscr{H}} u=\mathbb{E}^{\mathscr{H}}
Consider on the measure space \(\left([0, \infty), \mathscr{B}[0, \infty),\left.\lambda^{1}ight|_{[0, \infty)}ight)\) the filtration defined by \(\mathscr{A}_{n}:=\sigma([0,1),[0,2), \ldots,[n-1,
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and \(\mathscr{G} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra. Show that, in general,\[\int \mathbb{E}^{\mathscr{G}} u d \mu
Prove Corollaries 27.14 and 27.15 .Data from corollary 27.14Data from corollary 27.15 Corollary 27.14 (conditional Fatou's lemma) Let (X,A,) be a o-finite measure space, CA be a sub-o-algebra, such
Let \(\left(X, \mathscr{A}, \mathscr{A}_{n}, \muight)\) be a \(\sigma\)-finite filtered measure space and denote by \(\langle u, \phiangle\) the canonical dual pairing between \(u \in L^{p}\) and
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset L^{1}(\mathscr{A})\). Show that\[m_{1}:=u_{1}, \quad
(Continuation of Problem 27.15 ). If \(\int u_{1} d \mu=0\) and \(\mathbb{E}^{\mathscr{A}_{n}} u_{n+1}=0\), then \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is called a martingale difference sequence.
Doob decomposition. Let \(\left(X, \mathscr{A}, \mathscr{A}_{n}, \muight)\) be a \(\sigma\)-finite filtered measure space and let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a
Let \((\Omega, \mathscr{A}, P)\) be a probability space and let \(\left(X_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent identically distributed random variables such that
Let \(\left(X, \mathscr{A}, \mathscr{A}_{n}, \muight)\) be a finite filtered measure space. Let \(u_{n} \in L^{1}\left(\mathscr{A}_{n}ight)\) be a sequence of measurable functions such that
Prove the orthogonality relation for the Jacobi polynomials 28.1 .Data from jacobi polynmials 28.1 Jacobi polynomials (J.)) EN , B> -1 28.1
Use the Gram-Schmidt orthonormalization procedure to verify the formulae for the Chebyshev, Legendre, Laguerre and Hermite polynomials given in items 28.1 -28.5.
State and prove Theorem 28.6 and Corollary 28.8 for an arbitrary compact interval \([a, b]\).Data from theorem 28.6Data from corollary 28.8 Theorem 28.6 (Weierstra) Polynomials are dense in C[0, 1]
Prove the orthogonality relations (28.4) for the trigonometric system.Equation 28.4 (-T,T) cos(jx) cos(kx) dx = (-,) L sin (lx) sin(mx) dx = (-TT) 0, , 2, if lm, T, if l=m, cos(kx) sin(x) dx=0 for
(i) Show that for suitable constants \(c_{k}, s_{k}, \sigma_{k} \in \mathbb{R}\) and all \(n \in \mathbb{N}_{0}\)\[\cos ^{n} x=\sum_{k=0}^{n} c_{k} \cos (k x), \sin ^{2 n+1} x=\sum_{k=1}^{n} s_{k}
Use the formula \(\sin a-\sin b=2 \cos ((a+b) / 2) \sin ((a-b) / 2)\) to show that \(D_{N}(x) \sin (x / 2)=\frac{1}{2} \sin \left(N+\frac{1}{2}ight) x\). This proves (28.11).Equation 28.11 Dy(x)= sin
Find the Fourier series expansion for the function \(|\sin x|\).
Let \(u(x)=\mathbb{1}_{[0,1)}(x)\). Show that the Haar-Fourier series for \(u\) converges for all \(1 \leqslant p
Show that the Haar-Fourier series for \(u \in C_{c}\) converges uniformly for every \(x\) to \(u(x)\). Show that this remains true for functions \(u \in C_{\infty}\), i.e. the set of continuous
Extend Problem 28.9 to the Haar wavelet expansion.[ use Problem 28.9 and show that \(\left\|\mathbb{E}^{\mathscr{A}_{-N}^{\Delta}} uight\|_{\infty} ightarrow 0\) for all \(u \in
Let \(u(x)=\mathbb{1}_{[0,1 / 3)}(x)\). Prove that the Haar-Fourier series diverges at \(x=\frac{1}{3}\).[ verify that \(\lim \inf _{N ightarrow \infty} s_{N}\left(u, \frac{1}{3}ight)
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space, \(f \in \mathcal{M}(\mathscr{A})\) and \(1 \leqslant p(i) If \(f \in L^{p}(\mu)\), then \(\|f\|_{p}=\sup \left\{\int f g d \mu: g
Weak convergence in \(L^{p}\). Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and let \(p \in(1, \infty]\), \(q \in[1, \infty)\) be conjugate indices. We assume that
Lévy's continuity theorem. The steps below sketch a proof of the following theorem.Theorem (P. Lévy; continuity theorem). Let \(\left(\mu_{i}ight)_{i \in \mathbb{N}}\) be a sequence of finite
Bochner's theorem. A function \(\phi: \mathbb{R}^{n} ightarrow \mathbb{C}\) is positive semidefinite, if the matrices \(\left(\phi\left(\xi_{i}-\xi_{k}ight)ight)_{i, k=1}^{m}\) are positive hermitian
Let \((X, d)\) be a locally compact metric space. Write \(C_{\infty}(X):=\overline{C_{c}(X)}\) for the closure of \(C_{c}(X)\) with respect to the uniform norm. Show that(i) \(C_{\infty}(X)=\left\{u
Use Theorem 21.18 for a new proof of Lévy's continuity theorem (Problem 21.3 ).(i) Show (as in Problem 21.3 ) that \(\lim _{i ightarrow \infty} \int u d \mu_{i}\) exists for all \(u \in
Let \((X, d)\) be a locally compact metric space and \(\mu, \mu_{n} \in \mathfrak{M}_{\mathrm{r}}^{+}(X), \mu_{n} \stackrel{\mathrm{v}}{ightarrow} \mu\). Prove that\[\lim _{n} \int_{B} u d
Let \((X, \mathscr{A}, \mu)\) be a finite measure space and \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{M}(\mathscr{A})\). Prove that\[\lim _{k ightarrow \infty} \mu\left\{\sup _{n
Show that for a sequence \(\left(u_{n}ight)_{n \in \mathbb{N}}\) of measurable functions on a finite measure space\[\lim _{k ightarrow \infty} \mu\left\{\sup _{n \geqslant
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{M}(\mathscr{A})\). Show that \(u_{n} ightarrow u\) \((n ightarrow
Consider one-dimensional Lebesgue measure \(\lambda\) on \(([0,1], \mathscr{B}[0,1])\). Compare the convergence behaviour (a.e., \(\mathcal{L}^{p}\), in measure) of the following sequences:(i)
Let \(\left(u_{n}ight)_{n \in \mathbb{N}},\left(w_{n}ight)_{n \in \mathbb{N}}\) be two sequences of measurable functions on \((X, \mathscr{A}, \mu)\). Suppose that \(u_{n}
Let \((X, \mathscr{A}, \mu)\) be a measure space which is not \(\sigma\)-finite. Construct an example of a sequence \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{M}(\mathscr{A})\) which
(i) Prove, without using Vitali's convergence theorem, the following theorem.Theorem (bounded convergence). Let \((X, \mathscr{A}, \mu)\) be a measure space, let \(A \in \mathscr{A}\) be a set with
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. Define for two random variables \(\xi, \eta\)\[ho_{\mathbb{P}}(\xi, \eta):=\inf \{\epsilon>0: \mathbb{P}\{|\xi-\eta| \geqslant
Let \(\left(u_{n}ight)_{n \in \mathbb{N}}\) be a sequence of measurable functions on a \(\sigma\)-finite measure space \((X, \mathscr{A}, \mu)\) and assume that \(u_{n} \stackrel{\mu}{ightarrow}
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space. Suppose that \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) satisfies \(\lim _{n ightarrow \infty}
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{M}(\mathscr{A})\).(i) Let \(\left(x_{n}ight)_{n \in \mathbb{N}} \subset \mathbb{R}\). Show
Let \(\mathcal{F}\) and \(\mathcal{G}\) be two families of uniformly integrable functions on an arbitrary measure space \((X, \mathscr{A}, \mu)\). Show that the following statements hold.(i) Every
Assume that \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is uniformly integrable. Show that\[\lim _{k ightarrow \infty} \frac{1}{k} \int \max _{n \leqslant k} u_{n} d \mu=0\]
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. Adapt the proof of Theorem 22.9 to show that a sequence \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{1}(\mu)\) is
Let \((X, \mathscr{A}, \mu)\) be a finite measure space. Show that \(\mathcal{F} \subset \mathcal{L}^{1}(\mu)\) is uniformly integrable if, and only if, the series \(\sum_{n=1}^{\infty} n \mu\{n[
Let \(\left(f_{i}ight)_{i \in I}\) be a family of uniformly integrable functions and let \(\left(u_{i}ight)_{i \in I} \subset \mathcal{L}^{1}(\mu)\) be some further family such that
Let \((X, \mathscr{A}, \mu)\) be an arbitrary measure space. Show that a family \(\mathcal{F} \subset \mathcal{L}^{1}(\mu)\) is uniformly integrable if, and only if, the following condition
Let \((X, \mathscr{A}, \mu)\) be a finite measure space and let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a martingale. Set \(\mathscr{A}_{0}:=\{\emptyset, X\}\) and
Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a (sub, super)martingale and let \(\left(\mathscr{B}_{n}ight)_{n \in \mathbb{N}}\) and \(\left(\mathscr{C}_{n}ight)_{n \in
Completion (7). Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a submartingale and denote by \(\overline{\mathscr{A}}_{n}\) the completion of \(\mathscr{A}_{n}\). Then \(\left(u_{n},
Show that \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is a submartingale if, and only if, \(u_{n} \in \mathcal{L}^{1}\left(\mathscr{A}_{n}ight)\) for all \(n \in \mathbb{N}\) and\[\int_{A} u_{n} d \mu
Prove the assertion made in Remark 23.2 (ii).Data from remark 23.2 (ii) (ii) Set Sn = {AE An: (A)
Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a martingale with \(u_{n} \in \mathcal{L}^{2}\left(\mathscr{A}_{n}ight)\). Show that\[\int u_{n} u_{k} d \mu=\int u_{n \wedge k}^{2} d
Martingale transform. Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a martingale and let \(\left(f_{n}ight)_{n \in \mathbb{N}}\) be a sequence of bounded functions such that \(f_{n}
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\left(\xi_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent identically distributed random variables with \(\xi_{n}
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\left(\xi_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent random variables with \(\xi_{n} \in
Martingale difference sequence. Let \(\left(d_{i}ight)_{i \in \mathbb{N}}\) be a sequence in \(\mathcal{L}^{2}(\mathscr{A}) \cap \mathcal{L}^{1}(\mathscr{A})\). Define \(\mathscr{A}_{0}:=\{\emptyset,
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\left(\xi_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent identically Bernoulli \((p, 1-p)\)-distributed random
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space, let \(u\) be a further measure on \(\mathscr{A}\) and let \(\left(A_{n, i}ight)_{i \in \mathbb{N}} \subset \mathscr{A}\) be for
Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a supermartingale and \(u_{n} \geqslant 0\) a.e. Prove that \(u_{k}=0\) a.e. implies that \(u_{k+n}=0\) a.e. for all \(n \in
Verify that the family \(\mathscr{A}_{\tau}\) defined in Definition 23.5 is indeed a \(\sigma\)-algebra.Data from definition 23.5 Definition 23.5 Let (X, A, An, ) be a o-finite filtered measure
Show that \(\tau\) is a stopping time if, and only if, \(\{\tau=n\} \in \mathscr{A}_{n}\) for all \(n \in \mathbb{N}_{0}\).
Show that, in the notation of Lemma 23.6 , \(\mathscr{A}_{\sigma \wedge \tau}=\mathscr{A}_{\sigma} \cap \mathscr{A}_{\tau}\) for any two stopping times \(\sigma, \tau\).Data from lemma 23.6 Lemma
Verify that the random times \(\sigma_{k}\) and \(\tau_{k}\) defined in the proof of Lemma 24.1 are stopping times.Data from lemma 24.1 Lemma 24.1 (Doob's upcrossing estimate) Let (un)neN be a
Let \(\left(\mathscr{A}_{-n}ight)_{n \in \mathbb{N}}\) be a decreasing filtration such that \(\left.\muight|_{\mathscr{A}_{-\infty}}\) is \(\sigma\)-finite. Assume that \(\left(u_{-n},
Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a supermartingale such that \(u_{n} \geqslant 0\) and \(\lim _{n ightarrow \infty} \int u_{n} d \mu=0\). Then \(u_{n} ightarrow 0\)
Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a martingale. If \(\mathcal{L}^{1}-\lim _{n ightarrow \infty} u_{n}\) exists, then the pointwise limit \(\lim _{n ightarrow \infty}
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. Find a martingale \(\left(u_{n}ight)_{n \in \mathbb{N}}\) for which \(0
The following exercise furnishes an example of a martingale \(\left(M_{n}ight)_{n \in \mathbb{N}}\) on the probability space \(\left([0,1], \mathscr{B}[0,1],
Consider the probability space \((\mathbb{N}, \mathscr{P}(\mathbb{N}), \mathbb{P})\) with\[\mathbb{P}\{n\}:=\frac{1}{n}-\frac{1}{n+1}\]Set\[\mathscr{A}_{n}:=\sigma(\{1\},\{2\}, \ldots,\{n\},[n+1,
\(\mathcal{L}^{2}\)-bounded martingales. A martingale \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) is called \(\mathcal{L}^{2}\)-bounded, if the \(\mathcal{L}^{2}\)-norms are bounded:
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space.(i) Let \(\left(\varepsilon_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent identically Bernoulli \(\left(\frac{1}{2},
Show that Theorem 24.6 is enough to prove the Radon-Nikodým theorem (Theorem 25.2 ) for a countably generated \(\mathscr{A}\), i.e. \(\mathscr{A}=\sigma\left(\left\{A_{n}ight\}_{n \in
Let \(\mu\) and \(u\) be measures on the measurable space \((X, \mathscr{A})\). Show that the absolute continuity condition (25.1) is equivalent to\[\mu(A \triangle B)=0 \Longrightarrow u(A)=u(B)
A theorem of Doob. Let \(\left(\mu_{t}ight)_{t \geqslant 0}\) and \(\left(u_{t}ight)_{t \geqslant 0}\) be two families of measures on the \(\sigma\)-finite measure space \((X, \mathscr{A})\) such
A measure \(u\) on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)\) is called quasi-invariant if for all \(N \in \mathscr{B}\left(\mathbb{R}^{n}ight)\) with \(u(N)=0\) it holds
Do any of the problems at the end of Chapter 20.
Conditional expectations. Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and \(\mathscr{F} \subset \mathscr{A}\) be a sub\(\sigma\)-algebra. Then use the Radon-Nikodým theorem to
Let \(\mu, u\) be \(\sigma\)-finite measures on the measurable space \((X, \mathscr{A})\). Let \(\left(\mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a filtration of sub- \(\sigma\)-algebras of
Kolmogorov's inequality. Let \(\left(\xi_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent, identically distributed random variables on a probability space \((\Omega, \mathscr{A},
Let u,w⩾0u,w⩾0 be measurable functions on a σσ-finite measure space (X,A,μ)(X,A,μ).(i) Show that tμ{u⩾t}⩽∫{u⩾t}wdμtμ{u⩾t}⩽∫{u⩾t}wdμ for all t>0t>0 implies
Show the following improvement of Doob's maximal inequality Theorem 25.12 . Let \(\left(u_{n}ight)_{n \in \mathbb{N}}\) be a martingale or \(\left(\left|u_{n}ight|^{p}ight)_{n \in \mathbb{N}},
\(\mathcal{L}^{p}\)-bounded martingales. A martingale \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) on a \(\sigma\)-finite filtered measure space is called \(\mathcal{L}^{p}\)-bounded, if
Use Theorem 24.6 to show that the martingale of Example 25.14 is uniformly integrable.Data from example 25.14Data from theorem 24.6 Example 25.14 Consider in R" the half-open squares Qk (2):=z+[0,2
Let \(u:[a, b] ightarrow \mathbb{R}\) be a continuous function. Show that \(x \mapsto \int_{[a, x]} u(t) d t\) is everywhere differentiable and find its derivative. What happens if we assume only
Let \(f: \mathbb{R} ightarrow \mathbb{R}\) be a bounded increasing function. Show that \(f^{\prime}\) exists Lebesgue a.e. and \(f(b)--f(a) \geqslant \int_{(a, b)} f^{\prime}(x) d x\). When do we
Fubini's 'other' theorem. Let \(\left(f_{n}ight)_{n \in \mathbb{N}}\) be a sequence of monotone increasing functions \(f_{n}\) : \([a, b] ightarrow \mathbb{R}\). If the series

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