Questions and Answers of Measures Integrals And Martingales

Show that the examples given in Example 26.5 are indeed inner product spaces.Data from example 26.5 Example 26.5 (1) The typical finite-dimensional R" (R-vector space) n (x, y) = xiyi i=1 1/2 Ball=
This exercise shows the following theorem.Theorem (Fréchet-von Neumann-Jordan). An inner product \(\langle\cdot, \cdotangle\) on the \(\mathbb{R}\)-vector space \(V\) derives from a norm if, and
(Continuation of Problem 26.2 ) Assume now that \(W\) is a \(\mathbb{C}\)-vector space with norm \(\|\cdot\|\) satisfying the parallelogram identity (26.3) and let\[(v,
Does the norm \(\|\cdot\|_{1}\) on \(L^{1}\left([0,1], \mathscr{B}[0,1],\left.\lambda^{1}ight|_{[0,1]}ight)\) derive from an inner product?
Let \((V,\langle\cdot, \cdotangle)\) be a \(\mathbb{C}\)-inner product space, \(n \in \mathbb{N}\) and set \(\theta:=e^{2 \pi i / n}\).(i) Show that\[\frac{1}{n} \sum_{j=1}^{n} \theta^{j k}=
Let \(V\) be a real inner product space. Show that \(v \perp w\) if, and only if, Pythagoras' theorem \(\|v+w\|^{2}=\|v\|^{2}+\|w\|^{2}\) holds.
Show that every convergent sequence in \(\mathcal{H}\) is a Cauchy sequence.
Show that \(g \mapsto\langle g, hangle, h \in \mathcal{H}\), is continuous.
Show that \(\|(g, h)\|:=\left(\|g\|^{p}+\|h\|^{p}ight)^{1 / p}\) is for every \(p \geqslant 1\) a norm on \(\mathcal{H} \times \mathcal{H}\). For which values of \(p\) does \(\mathcal{H} \times
Show that \((g, h) \mapsto\langle g, hangle\) and \((t, h) \mapsto t h\) are continuous on \(\mathcal{H} \times \mathcal{H}\), resp. \(\mathbb{R} \times \mathcal{H}\).
Show that a Hilbert space \(\mathcal{H}\) is separable if, and only if, \(\mathcal{H}\) contains a countable maximal orthonormal system.
Let \(w \in \mathcal{H}=L^{2}(X, \mathscr{A}, \mu)\) and show that \(M_{w}^{\perp}:=\left\{u \in L^{2}: \int u w d \mu=0ight\}^{\perp}\) is either \(\{0\}\) or a one-dimensional subspace of
Let \(\left(e_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{H}\) be an orthonormal system.(i) Show that no subsequence of \(\left(e_{n}ight)_{n \in \mathbb{N}}\) converges. However, for every \(h \in
Let \(\mathcal{H}\) be a real Hilbert space.(i) Show that\[\|h\|=\sup _{g eq 0} \frac{|\langle g, hangle|}{\|g\|}=\sup _{\|g\| \leqslant 1}|\langle g, hangle|=\sup _{\|g\|=1}|\langle g, hangle|\](ii)
Show that the linear span of a sequence \(\left(e_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{H}, \operatorname{span}\left\{e_{n}: e_{n} \in \mathcal{H}, n \in \mathbb{N}ight\}\), is a linear
A weak form of the uniform boundedness principle. Consider the real Hilbert space \(\ell^{2}=\ell_{\mathbb{R}}^{2}(\mathbb{N})\) and let \(a=\left(a_{n}ight)_{n \in \mathbb{N}}\) and
Let \(F, G \subset \mathcal{H}\) be linear subspaces. An operator \(P\) defined on \(G\) is called ( \(\mathbb{K}\)-)linear if \(P(\alpha f+\beta g)=\alpha P f+\beta P g\) holds for all \(\alpha,
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) be mutually disjoint sets such that \(X=\biguplus_{n \in \mathbb{N}} A_{n}\).
Let \((X, \mathscr{A}, \mu)\) be a measure space and assume that \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) is a sequence of pairwise disjoint sets such that \(\bigcup_{n \in
Stieltjes measure (1).(i) Let \(\mu\) be a measure on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\) such that \(\mu[-n, n)0 \\ 0 & \text { if } x=0 \\ -\mu[x, 0) & \text { if } x
Let \((X, \mathscr{A}, \mu)\) be a finite measure space, \(\mathscr{B} \subset \mathscr{A}\) a Boolean algebra (i.e. \(X \in \mathscr{B}, \mathscr{B}\) is stable under the formation of finite unions,
Let \((X, \mathscr{A}, \mu)\) be a measure space such that all singletons \(\{x\} \in \mathscr{A}\). A point \(x\) is called an atom, if \(\mu\{x\}>0\). A measure is called non-atomic or diffuse, if
A set \(A \subset \mathbb{R}^{n}\) is called bounded if it can be contained in a ball \(B_{r}(0) \supset A\) of finite radius \(r\). A set \(A \subset \mathbb{R}^{n}\) is called pathwise connected if
Show that every continuous function \(u: \mathbb{R} ightarrow \mathbb{R}\) is \(\mathscr{B}(\mathbb{R}) / \mathscr{B}(\mathbb{R})\)-measurable.[ check that for continuous functions \(\{u>\alpha\}\)
Consider \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\) and \(u: \mathbb{R} ightarrow \mathbb{R}\). Show that \(\{x\} \in \sigma(u)\) for all \(x \in \mathbb{R}\) if, and only if, \(u\) is injective.
Let \((\Omega, \mathscr{A})\) be a measurable space and \(\xi: \mathbb{R} \times \Omega ightarrow \mathbb{R}\) be a map such that \(\omega \mapsto \xi(t, \omega)\) is \(\mathscr{A} /
Let \((X, \mathscr{A}, \mu)\) be a measure space and \((X, \overline{\mathscr{A}}, \bar{\mu})\) its completion (see Problem 4.15 ). Show that a function \(\phi: X ightarrow \mathbb{R}\) is
Let \(u \in \mathcal{L}^{1}(0,1)\) be positive and monotone. Find the limit\[\lim _{n ightarrow \infty} \int_{0}^{1} u\left(t^{n}ight) d t\]
Let \(u \in \mathcal{L}^{1}(0,1)\). Find the limit\[\lim _{n ightarrow \infty} \int_{0}^{1} t^{n} f(t) d t\]
Show that: [Use the geometric series to express (e−t−1)−1(e−t−1)−1, observe that sint=Imeitsin⁡t=Im⁡eit and use Problem 10.9 .]Data from problem 10.9 sin t el dt -; H=I 1 n? +1
Let \(u: \mathbb{R} ightarrow \mathbb{R}\) be a Borel measurable function and assume that \(x \mapsto e^{\lambda x} u(x)\) is integrable for each \(\lambda \in \mathbb{R}\). Show that for all \(z \in
Let λλ be one-dimensional Lebesgue measure. Show that for every integrable function uu, the integral function or primitiveis continuous. What happens if we exchange λλ for a general measure μμ
Consider the functions(i) \(u(x)=\frac{1}{x}, \quad x \in[1, \infty)\);(ii) \(v(x)=\frac{1}{x^{2}}, \quad x \in[1, \infty)\);(iii) \(\quad w(x)=\frac{1}{\sqrt{x}}, \quad x \in(0,1]\)(iv)
Show that the function \(\mathbb{R} i x \mapsto \exp \left(-x^{\alpha}ight)\) is \(\lambda^{1}(d x)\)-integrable over the set \([0, \infty)\) for every \(\alpha>0\).[find integrable majorants \(u\)
Show that for every parameter α>0α>0 the functionx↦(sinxx)3e−αxx↦(sin⁡xx)3e−αxis integrable over (0,∞)(0,∞) and that the integral is continuous as a function of the parameter.
Show that the function\[G: \mathbb{R} ightarrow \mathbb{R}, \quad G(x):=\int_{\mathbb{R} \backslash\{0\}} \frac{\sin (t x)}{t\left(1+t^{2}ight)} d t\]is differentiable and find \(G(0)\) and
Denote by \(\lambda\) one-dimensional Lebesgue measure. Prove that(i) \(\int_{(1, \infty)} e^{-x} \ln (x) \lambda(d x)=\lim _{k ightarrow \infty} \int_{(1, k)}\left(1-\frac{x}{k}ight)^{k} \ln (x)
Denote by \(\lambda\) Lebesgue measure on \(\mathbb{R}\) and set\[F(t):=\int_{(0, \infty)} e^{-x} \frac{t}{t^{2}+x^{2}} \lambda(d x), \quad t>0\]Show that \(F(0+)=\lim _{t \downarrow 0} F(t)=\pi /
Let \(\mu\) be a measure on \((\mathbb{R}, \mathscr{B}(\mathbb{R})\) ), let \(u: \mathbb{R} ightarrow \mathbb{C}\) be a measurable function (see Problem 10.9 ) and denote by \(d x\) one-dimensional
Let \(\phi \in \mathcal{L}^{1}([0,1], d x)\) and define \(f(t):=\int_{[0,1]}|\phi(x)-t| d x\). Show that(i) \(f\) is continuous,(ii) \(f\) is differentiable at \(t \in \mathbb{R}\) if, and only if,
Let f(t):=∫∞0x−2sin2xe−txdx,t⩾0f(t):=∫0∞x−2sin2⁡xe−txdx,t⩾0.(i) Show that ff is continuous on [0,∞)[0,∞) and twice differentiable on (0,∞)(0,∞).(iii) Use (i) and (ii) to
Show that \(\int_{0}^{\infty} x^{n} e^{-x} d x=n\) ! for all \(n \in \mathbb{N}\).[ Show that \(\int_{0}^{\infty} e^{-x t} d x=1 / t, t>0\), and differentiate this identity.][ Show that
Euler's gamma function. Show that the function\[\Gamma(t):=\int_{(0, \infty)} e^{-x} x^{t-1} d x, \quad t>0\]has the following properties.(i) It is \(m\)-times differentiable with
Denote by \(\lambda\) one-dimensional Lebesgue measure on the interval \((0,1)\).(i) Show that for all \(k \in \mathbb{N}_{0}\) one has\[\int_{(0,1)}(x \ln x)^{k} \lambda(d
Show that \(x \mapsto x^{n} f(u, x), f(u, x)=e^{u x} /\left(e^{x}+1ight), 0
Calculate the following limit:\[\lim _{n ightarrow \infty} \int_{0}^{1} \frac{1+n x^{2}}{\left(1+x^{2}ight)^{n}} d x\]
Moment generating function. Let \(X\) be a positive random variable on the probability space \((\Omega, \mathscr{A}, \mathbb{P})\). The function \(\phi_{X}(t):=\int e^{-t X} d \mathbb{P}\) is called
Consider the functions \(u(x)=\mathbb{1}_{\mathbb{Q} \cap[0,1]}\) and \(v(x)=\mathbb{1}_{\left\{n^{-1}: n \in \mathbb{N}ight\}}(x)\). Prove or disprove the following statements.(i) The function \(u\)
Construct a sequence of functions \(\left(u_{n}ight)_{n \in \mathbb{N}}\) which are Riemann integrable but converge to a limit \(u_{n} ightarrow u\) which is not Riemann integrable.[ consider e.g.
Assume that \(u:[0, \infty) ightarrow \mathbb{R}\) is positive and improperly Riemann integrable. Show that \(u\) is also Lebesgue integrable.
Fresnel integrals. Show that the following improper Riemann integrals exist:\[\int_{0}^{\infty} \sin x^{2} d x \text { and } \int_{0}^{\infty} \cos x^{2} d x\]Do they exist as Lebesgue
Frullani's integral. Let f:(0,∞)→Rf:(0,∞)→R be a continuous function such that limx→0f(x)=mlimx→0f(x)=m and limx→∞f(x)=Mlimx→∞f(x)=M. Show that the two-sided improper Riemann
Let \((X, \mathscr{A}, \mu)\) be a finite measure space and let \(1 \leqslant q
Let \((X, \mathscr{A}, \mu)\) be a general measure space and \(1 \leqslant p \leqslant r \leqslant q \leqslant \infty\).Prove that \(\mathcal{L}^{p}(\mu) \cap \mathcal{L}^{q}(\mu) \subset
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(u, v \in \mathcal{L}^{p}(\mu)\).(i) Find conditions which guarantee that \(u v, u+v\) and \(\alpha u, \alpha \in \mathbb{R}\) are in
Let \(\Omega\) be a set and \(B, B^{c} \subset \Omega\) such that \(B\) and \(B^{c}\) are not empty.(i) Find all measurable functions \(u:(\Omega,\{\emptyset, \Omega\}) ightarrow(\mathbb{R},
Generalized Hölder inequality. Iterate Hölder's inequality to derive the following generalization:\[\int\left|u_{1} \cdot u_{2} \cdots u_{N}ight| d \mu \leqslant\left\|u_{1}ight\|_{p_{1}}
Young functions. Let \(\phi:[0, \infty) ightarrow[0, \infty)\) be a strictly increasing continuous function such that \(\phi(0)=0\) and \(\lim _{\xi ightarrow \infty} \phi(\xi)=\infty\). Denote by
Let \(1 \leqslant p
Consider one-dimensional Lebesgue measure on \([0,1]\). Verify that the sequence \(u_{n}(x):=\) \(n 1_{(0,1 / n)}(x), n \in \mathbb{N}\), converges pointwise to the function \(u \equiv 0\), but that
Let \(p, q \in[1, \infty]\) be conjugate, i.e. \(p^{-1}+q^{-1}=1\), and assume that \(\left(u_{k}ight)_{k \in \mathbb{N}} \subset \mathcal{L}^{p}\) and \(\left(w_{k}ight)_{k \in \mathbb{N}} \subset
Prove that \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{2}(\mu)\) converges in \(\mathcal{L}^{2}\) if, and only if, \(\lim _{n, m ightarrow \infty} \int u_{n} u_{m} d \mu\) exists.[
Let \((X, \mathscr{A}, \mu)\) be a finite measure space. Show that every measurable \(u \geqslant 0\) with \(\int \exp (h u(x)) \mu(d x)0\) is in \(\mathcal{L}^{p}(\mu)\) for every \(p \geqslant
Let \(\lambda\) be Lebesgue measure in \((0, \infty)\) and \(p, q \geqslant 1\) arbitrary.(i) Show that \(u_{n}(x):=n^{\alpha}(x+n)^{-\beta}(\alpha \in \mathbb{R}, \beta>1)\) holds for every \(n \in
Let \(u(x)=\left(x^{\alpha}+x^{\beta}ight)^{-1}, x, \alpha, \beta>0\). For which \(p \geqslant 1\) is \(u \in \mathcal{L}^{p}\left(\lambda^{1},(0, \infty)ight)\) ?
Consider the measure space \((\Omega=\{1,2, \ldots, n\}, \mathscr{P}(\Omega), \mu), n \geqslant 2\), where \(\mu\) is the counting measure. Show that
Let \((X, \mathscr{A}, \mu)\) be a measure space. The space \(\mathcal{L}^{p}(\mu)\) is called separable if there exists a countable dense subset \(\mathscr{D}_{p} \subset \mathcal{L}^{p}(\mu)\).
Let \(u_{n} \in \mathcal{L}^{p}, p \geqslant 1\), for all \(n \in \mathbb{N}\). What can you say about \(u\) and \(w\) if you know that \(\lim _{n ightarrow \infty} \int\left|u_{n}-uight|^{p} d
Let \((X, \mathscr{A}, \mu)\) be a finite measure space and let \(u \in \mathcal{L}^{1}(\mu)\) be strictly positive with \(\int u d \mu=1\). Show that\[\int(\log u) d \mu \leqslant \mu(X) \log
Let \(u\) be a positive measurable function on \([0,1]\). Which of the following is larger:\[\int_{(0,1)} u(x) \log u(x) \lambda(d x) \quad \text { or } \quad \int_{(0,1)} u(s) \lambda(d s) \cdot
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(p \in(0,1)\). The conjugate index is given by \(q:=\) \(p /(p-1)
Let \((X, \mathscr{A}, \mu)\) be a finite measure space and \(u \in \mathcal{M}(\mathscr{A})\) be a bounded function with \(\|u\|_{\infty}>0\). Prove that for all \(n \in \mathbb{N}\)(i)
Let \((X, \mathscr{A}, \mu)\) be a general measure space and let \(u \in \bigcap_{p \geqslant 1} \mathcal{L}^{p}(\mu)\). Then\[\lim _{p ightarrow \infty}\|u\|_{p}=\|u\|_{\infty}\]where
Let \((X, \mathscr{A}, \mu)\) be a probability space and assume that \(\|u\|_{q}0\). Show that \(\lim _{p ightarrow 0}\|u\|_{p}=\exp \left(\int \log |u| d \muight)\) (we set \(e^{-\infty}:=0\) ).
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(1 \leqslant p
Variants of Jensen's inequality. Let \((X, \mathscr{A}, \mu)\) be a probability space.(i) Show Jensen's inequality for convex \(V: \mathbb{R} ightarrow \mathbb{R}\), see Example 13.14 (v).(ii) Show
Use Jensen's inequality (Example 13.14 (i), (ii)) to derive Hölder's inequality and Minkowski's inequality. Use\[\Lambda(x)=x^{1 / q}, x \geqslant 0, \quad w=|f|^{p} \quad \text { and } \quad
Let \((X, \mathscr{A}, \mu)\) be a finite measure space, \(1 \leqslant p
Prove the rules (14.2) for Cartesian products.Equation 14.2 (U4) x B=U(A B), (NA) x B=n(A B), (A x B) n (A' x B') = (ANA') x (BnB'), A x B= (X x B) \ (A x B), AXBCA' XB'ACA' und BCB', (14.2)
Let \((X, \mathscr{A}, \mu)\) and \((Y, \mathscr{B}, u)\) be two \(\sigma\)-finite measure spaces. Show that \(A \times N\), where \(A \in \mathscr{A}\) and \(N \in \mathscr{B}, u(N)=0\), is a \(\mu
Let \(\left(X_{i}, \mathscr{A}_{i}, \mu_{i}ight), i=1,2\), be \(\sigma\)-finite measure spaces and \(f: X_{1} \times X_{2} ightarrow \mathbb{C}\) a measurable function. A function is negligible
Denote by \(\lambda\) Lebesgue measure on \((0, \infty)\). Prove that the following iterated integrals exist and that\[\int_{(0, \infty)} \int_{(0, \infty)} e^{-x y} \sin x \sin y \lambda(d x)
Denote by \(\lambda\) Lebesgue measure on \((0,1)\). Show that the following iterated integrals exist, but yield different values:\[\int_{(0,1)} \int_{(0,1)}
Denote by \(\lambda\) Lebesgue measure on \((-1,1)\). Show that the iterated integrals exist and coincide,\[\int_{(-1,1)} \int_{(-1,1)} \frac{x y}{\left(x^{2}+y^{2}ight)^{2}} \lambda(d x) \lambda(d
Evaluate \(\int_{0}^{1} \int_{0}^{1} f(x, y) d x d y, \int_{0}^{1} \int_{0}^{1} f(x, y) d y d x\) and \(\int_{[0,1]^{2}}|f(x, y)| d(x, y)\) if(a) \(\left(x-\frac{1}{2}ight)^{-3} \mathbb{1}_{\left\{0
Consider the measure space \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\) and denote by \(\zeta_{M}: \mathscr{B}(\mathbb{R}) ightarrow[0, \infty], M \subset \mathbb{R}\), \(\zeta_{M}(A):=\#(A \cap M)\)
(i) Evaluate\[\int_{[0, \infty)^{2}} \frac{d x d y}{(1+y)\left(1+x^{2} yight)}\](ii) Use (i), in order to evaluate\[\int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x\](iii) Use a series representation in
Let \(\mu, u\) be \(\sigma\)-finite measures on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\). Show that(i) The set \(D:=\{x \in \mathbb{R}: \mu\{x\}>0\}\) is at most countable.(ii) The diagonal \(\Delta
Let \(\mu(A):=\# A\) be the counting measure and \(\lambda\) be Lebesgue measure on the measurable space \(([0,1], \mathscr{B}[0,1])\). Denote by \(\Delta:=\left\{(x, y) \in[0,1]^{2}: x=yight\}\) the
(i) State Tonelli's and Fubini's theorems for spaces of sequences, i.e. for the measure space \((\mathbb{N}, \mathscr{P}(\mathbb{N}), \mu)\), where \(\mu:=\sum_{n \in \mathbb{N}} \delta_{n}\), and
Let \(u: \mathbb{R}^{2} ightarrow[0, \infty]\) be a measurable function. \(S[u]:=\{(x, y): 0 \leqslant y \leqslant u(x)\}\) is the set above the abscissa and below the graph \(\Gamma[u]:=\{(x, u(x)):
Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and let \(u \in \mathcal{M}^{+}(\mathscr{A})\) be a \([0, \infty]\)-valued measurable function. Show that the set\[Y:=\{y \in
Completion (5). Let \((X, \mathscr{A}, \mu)\) and \((Y, \mathscr{B}, u)\) be any two measure spaces such that \(\mathscr{A} eq\) \(\mathscr{P}(X)\) and such that \(\mathscr{B}\) contains non-empty
Let \(\mu\) be a bounded measure on the measure space \(([0, \infty), \mathscr{B}[0, \infty))\).(i) Show that \(A \in \mathscr{B}[0, \infty) \otimes \mathscr{P}(\mathbb{N})\) if, and only if,
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space, i.e. a measure space such that \(\mathbb{P}(\Omega)=1\). Show that for a measurable function \(T: \Omega ightarrow[0, \infty)\) and
Stieltjes measure (2). Stieltjes integrals. This continues Problem 6.1. Let \(\mu\) and \(u\) be two measures on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\) such that \(\mu((-n, n]), u((-n, n])the
Rearrangements. Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and let \(f \in \mathcal{L}^{p}(\mu)\) for some \(p \in[1, \infty)\). The distribution function of \(f\) is given by
The differentiability lemma revisited. Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and \(\phi\) : \(\mathbb{R} \times X ightarrow \mathbb{R}\) a mapping with the following
Let \((X, \mathscr{A}, \mu)\) be a measure space and let \(T: X ightarrow X\) be a bijective measurable map whose inverse \(T^{-1}: X ightarrow X\) is again measurable. Show that for every \(f \in
Let \(u \in \mathcal{L}^{1}\left(\mathbb{R}^{n}, \lambda^{n}ight)\) and \(\epsilon>0\). Show that \(\int u(\epsilon x) d x=\epsilon^{-n} \int u(y) d y\).

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