Questions and Answers of Measures Integrals And Martingales

Let \(u: \mathbb{R} ightarrow \mathbb{R}\) be measurable. Which of the following functions are measurable:\(u(x-2)\)\(e^{u(x)}\),\(\sin (u(x)+8)\),\(u^{\prime \prime}(x)\),\(\operatorname{sgn} u(x-7)
Check the following. In the proof of the factorization lemma (Lemma 8.14 ) we cannot, in general, replace \(\liminf _{n} w_{n}\) by \(\lim _{n} w_{n}\).[ find a sequence \(\left(w_{n}ight)_{n}\) and
One can show that there are non-Borel measurable sets \(A \subset \mathbb{R}\), see Appendix \(\mathrm{G}\). Taking this fact for granted, show that measurability of \(|u|\) does not, in general,
Show that every increasing function \(u: \mathbb{R} ightarrow \mathbb{R}\) is \(\mathscr{B}(\mathbb{R}) / \mathscr{B}(\mathbb{R})\)-measurable. Under which additional condition(s) do we have
Let \(\mathscr{A}=\sigma(\mathscr{G})\) be a \(\sigma\)-algebra on \(X\), where \(\mathscr{G}=\left\{G_{i}: i \in \mathbb{N}ight\}\) is a countable generator. Let \(g:=\sum_{i=1}^{\infty} 2^{-i}
Show that any left- or right-continuous function \(u: \mathbb{R} ightarrow \mathbb{R}\) is Borel measurable.
Show that every linear map \(f: \mathbb{R}^{n} ightarrow \mathbb{R}^{m}\) is \(\mathscr{B}\left(\mathbb{R}^{n}ight) / \mathscr{B}\left(\mathbb{R}^{m}ight)\)-measurable. Provide an example in which
Complete the proof of Properties 9.8.Data from properties 9.8 (of the integral) Let u, v M(). Then (1) [1 d= (4) (audu-adp VAEA; Va>0 (iii) [(u + v) d = [ud + + [vd Sud < [vd (iv) u
Let \(f: X ightarrow \mathbb{R}\) be a positive simple function of the form \(f(x)=\sum_{n=1}^{m} \xi_{n} \mathbb{1}_{A_{n}}(x), \xi_{n} \geqslant 0\), \(A_{n} \in \mathscr{A}\) - but not necessarily
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(A_{1}, \ldots, A_{N} \in \mathscr{A}\) such that \(\mu\left(A_{n}ight)
Find an example showing that an 'increasing sequence of functions' is, in general, different from a 'sequence of increasing functions'.
Let \((X, \mathscr{A}, \mu)\) be a measure space. Show the following variant of Theorem 9.6. If \(u_{n} \geqslant 0\) are measurable functions such that for some \(u\) we have\[\exists K \in
Complete the proof of Corollary 9.9 and show that (9.6) is actually equivalent to (9.5) in Beppo Levi's theorem.Data from corollary 9.9Data from theorem 9.6 18 Let (Un)neNCM(A). Then 1 un is
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(u \in \mathcal{M}^{+}(\mathscr{A})\). Show that \(A \mapsto \int \mathbb{1}_{A} u d \mu, A \in \mathscr{A}\), is a measure.
Prove that every function \(u: \mathbb{N} ightarrow \mathbb{R}\) on \((\mathbb{N}, \mathscr{P}(\mathbb{N}))\) is measurable.
Let \((X, \mathscr{A})\) be a measurable space and \(\left(\mu_{n}ight)_{n \in \mathbb{N}}\) be a sequence of measures thereon. Set, as in Example 9.10(ii), \(\mu=\sum_{n \in \mathbb{N}} \mu_{n}\).
Reverse Fatou lemma. Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{M}^{+}(\mathscr{A})\). If \(u_{n} \leqslant u\) for all \(n \in
Let \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) be a sequence of disjoint sets such that \(\bigcup_{n \in \mathbb{N}} A_{n}=X\). Show that for every \(u \in
Kernels. Let \((X, \mathscr{A}, \mu)\) be a measure space. A map \(N: X \times \mathscr{A} ightarrow[0, \infty]\) is called a kernel if\[\begin{array}{rlrl}A & \mapsto N(x, A) & & \text { is a
(Continuation of Problem 6.3) Consider on \(\mathbb{R}\) the \(\sigma\)-algebra \(\Sigma\) of all Borel sets which are symmetric w.r.t. the origin. Set \(A^{+}:=A \cap[0, \infty), A^{-}:=(-\infty,
Prove Remark 10.5, i.e. prove the linearity of the integral.Data from remark 10.5 If u(x) = v(x) is defined in R for all x EX-i.e. if we can exclude '-' - then Theorem 10.4(i), (ii) just say that the
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. Find a counterexample to the claim that every \(\mathbb{P}\)-integrable function \(u \in \mathcal{L}^{1}(\mathbb{P})\) is bounded.[
Prove Lemma 10.8.Data from lemma 10.8 On the measure space (X, A,p) let us M(A). The set function + 1 ud = [1 sudu, AE A V: A+ is a measure on (X, A). It is called the measure with density (function)
Let \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) be a sequence of mutually disjoint sets. Show that\[u \mathbb{1}_{\cup_{n} A_{n}} \in \mathcal{L}^{1}(\mu) \Longleftrightarrow u
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(u \in \mathcal{M}(\mathscr{A})\). Show that\[u \in \mathcal{L}^{1}(\mu) \Longleftrightarrow \sum_{n \in \mathbb{Z}} 2^{n} \mu\left\{2^{n}
Let \((X, \mathscr{A}, \mu)\) be a finite measure space. Show that(i) \(u \in \mathcal{L}^{1}(\mu) \Longleftrightarrow u \in \mathcal{M}(\mathscr{A})\) and \(\sum_{n=0}^{\infty} \mu\{|u| \geqslant n\}
Generalized Fatou lemma. Assume that \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{1}(\mu)\). Prove the following.(i) If \(u_{n} \geqslant v\) for all \(n \in \mathbb{N}\) and some \(v
Independence (2). Let \((\Omega, \mathscr{A}, P)\) be a probability space and assume that the \(\sigma\)-algebras \(\mathscr{B}, \mathscr{C} \subset \mathscr{A}\) are independent (see Problem 5.11).
Integrating complex functions. Let \((X, \mathscr{A}, \mu)\) be a measure space. For a complex number \(z=x+i y \in \mathbb{C}\) we write \(x=\operatorname{Re} z\) and \(y=\operatorname{Im} z\) for
True or false: if \(f \in \mathcal{L}^{1}\) we can change \(f\) on a set \(N\) of measure zero (e.g. by\[\tilde{f}(x):= \begin{cases}f(x) & \text { if } x otin N \\ \beta & \text { if } x \in
Every countable set is a \(\lambda^{1}\)-null set. Use the Cantor ternary set \(C\) (see Problem 7.12) to illustrate that the converse is not true. What happens if we change \(\lambda^{1}\) to
Prove the following variants of the Markov inequality Proposition 11.5 . For all \(\alpha, c>0\) and whenever the expressions involved make sense/are finite,(i) \(\mu\{|u|>c\} \leqslant
Show that \(\int|u|^{p} d \mu
Completion (3). Let \((X, \overline{\mathscr{A}}, \bar{\mu})\) be the completion of \((X, \mathscr{A}, \mu)\), see Problems 4.15 , and 6.4.(i) Show that for every \(f^{*} \in
Completion (4). Inner measure and outer measure. Let \((X, \mathscr{A}, \mu)\) be a finite measure space. Define for every \(E \subset X\) the outer resp. inner measure\[\begin{aligned}&
Let \((X, \mathscr{A}, \mu)\) be a measure space and assume that \(u \in \mathcal{M}(\mathscr{A})\) and \(u=w\) almost everywhere w.r.t. \(\mu\). When can we say that \(w \in
'a.e.' is a tricky business. When working with 'a.e.' properties one has to be extremely careful. For example, the assertions ' \(u\) is continuous a.e.' and ' \(u\) is a.e. equal to an (everywhere)
Let \(\mu\) be a \(\sigma\)-finite measure on the measurable space \((X, \mathscr{A})\). Show that there exists a finite measure \(P\) on \((X, \mathscr{A})\) such that
Construct an example showing that for \(u, w \in \mathcal{M}^{+}(\mathscr{B})\) the equality \(\int_{B} u d \mu=\int_{B} w d \mu\) for all \(B \in \mathscr{B}\) does not necessarily imply that
Show the following extension of Corollary 11.7 . Let \(\mathscr{C} \subset \mathscr{P}(X)\) be a \(\cap\)-stable generator of \(\mathscr{A}\) which contains a sequence \(C_{n} \uparrow X\) such that
Egorov's theorem. Let \((X, \mathscr{A}, \mu)\) be a finite measure space and \(f_{n}: X ightarrow \mathbb{R}, n \in \mathbb{N}\), a sequence of measurable functions. Prove the following
Adapt the proof of Theorem 12.2 to show that any sequence (un)n∈N⊂M(A)(un)n∈N⊂M(A) with limn→∞un(x)=u(x)limn→∞un(x)=u(x) and |un|⩽g|un|⩽g for some g⩾0g⩾0 with
Give an alternative proof of Theorem 12.2 (ii) using the generalized Fatou theorem from Problem 10.7.Data from theorem 12.2Data from problem 10.7 (Lebesgue; dominated convergence) Let (X, A, u) be a
Prove the following result of W.H. Young [60]; among statisticians it is also known as Pratt's lemma, see J. W. Pratt [38].Theorem (Young; Pratt). Let (fk)k,(gk)k(fk)k,(gk)k and (Gk)k(Gk)k be
Let (un)n∈N(un)n∈N be a sequence of integrable functions on (X,A,μ)(X,A,μ). Show that, if ∑∞n=1∫|un|dμ∞∑n=1∞∫|un|dμ∞, the series ∑∞n=1un∑n=1∞un converges a.e. to a
Let \(\left(u_{n}\right)_{n \in \mathbb{N}}\) be a sequence of positive integrable functions on a measure space \((X, \mathscr{A}, \mu)\). Assume that the sequence decreases to \(0: u_{1} \geqslant
Find a sequence of integrable functions (un)n∈N(un)n∈N with un(x)→u(x)un(x)→u(x) for all xx and an integrable function uu but such that
Let \(\mu\) be a finite measure on \(([0, \infty), \mathscr{B}[0, \infty))\). Find the limit \(\lim _{r\rightarrow \infty} \int_{[0, \infty)} e^{-r x} \mu(d x)\).
Let \(\lambda\) denote Lebesgue measure on \(\mathbb{R}^{n}\).(i) Let \(u \in \mathcal{L}^{1}(\lambda)\) and \(K \subset \mathbb{R}^{n}\) be a compact (i.e. closed and bounded) set. Show that \(\lim
Let \(\lambda\) denote Lebesgue measure on \(\mathbb{R}^{n}\) and \(u \in \mathcal{L}^{1}(\lambda)\).(i) For every \(\epsilon>0\) there is a set \(B \in \mathscr{B}\left(\mathbb{R}^{n}ight),
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{1}(\mu)\) be a uniformly convergent sequence.(i) If \(\mu(X)
Consider the following variation of Cantor's set: fix \(r \in(0,1)\) and delete from \(I_{0}=[0,1]\) the open interval \(\left(\frac{1}{2}-\frac{1}{4} r, \frac{1}{2}+\frac{1}{4} ight)\). This defines
Let \(A, B, C \subset X\). The symmetric difference of \(A\) and \(B\) is \(A \triangle B:=(A \backslash B) \cup(B \backslash A)\). Verify that\[(A \cup B \cup C) \backslash(A \cap B \cap C)=(A
Show that the following sets have the same cardinality as \(\mathbb{N}:\{m \in \mathbb{N}: m\) is odd \(\}, \mathbb{N} \times \mathbb{Z}\), \(\mathbb{Q}^{m}(m \in \mathbb{N}), \bigcup_{m \in
Use Theorem 2.7 to show that \(\# \mathbb{N} \times \mathbb{N}=\# \mathbb{N}\).[ \(\# \mathbb{N}=\# \mathbb{N} \times\{1\}\) and \(\mathbb{N} \times\{1\} \subset \mathbb{N} \times \mathbb{N}\).
Show that if \(E \subset F\) we have \(\# E \leqslant \# F\). In particular, subsets of countable sets are again countable.
Let \(\mathscr{F}:=\{F \subset \mathbb{N}: \# F
Show - not using Theorem 2.10 - that \(\# \mathscr{P}(\mathbb{N})>\# \mathbb{N}\). Conclude that there are more than countably many maps \(f: \mathbb{N} ightarrow \mathbb{N}\).[ use the diagonal
If \(A \subset \mathbb{N}\) we can identify the indicator function \(\mathbb{1}_{A}: \mathbb{N} ightarrow\{0,1\}\) with the 0 -1-sequence \(\left(\mathbb{1}_{A}(i)ight)_{i \in \mathbb{N}}\), i.e.
Show that for \(A_{n}^{0}, A_{n}^{1} \subset X, n \in \mathbb{N}\), we have\[\bigcup_{n \in \mathbb{N}}\left(A_{n}^{0} \cap A_{n}^{1}ight)=\bigcap_{i=(i(k))_{k \in \mathbb{N}}
Let \(\mathscr{A}\) be a \(\sigma\)-algebra. Show that(i). if \(A_{1}, A_{2}, \ldots, A_{N} \in \mathscr{A}\), then \(A_{1} \cap A_{2} \cap \cdots \cap A_{N} \in \mathscr{A}\);(ii). \(A \in
Prove the assertions made in Example 3.3 (iv), (vi) and (vii). [ use (2.6) for (vii).]Data from example 3.3Equation 2.6 (i) P(X) is a o-algebra (the maximal o-algebra in X). (ii) {0, X) is a
Let \(X=\mathbb{R}\). Find the \(\sigma\)-algebra generated by the singletons \(\{\{x\}: x \in \mathbb{R}\}\).
Verify the assertions made in Remark 3.5 .Data from remark 3.5 Remark 3.5 (i) If is a o-algebra, then 9 = o(9). (ii) For ACX we have o({4})= {0, A, A, X}. (iii) If FCICA, then o(F) Co (9) Co(A) 3.5)
Let \(X=[0,1]\). Find the \(\sigma\)-algebra generated by the sets(i). \(\left(0, \frac{1}{2}ight)\);(ii). \(\left[0, \frac{1}{4}ight),\left(\frac{3}{4}, 1ight]\);(iii). \(\left[0,
Let \((X, \mathscr{A})\) be a measurable space. Show that there cannot be a \(\sigma\)-algebra \(\mathscr{A}\) which contains countably infinitely many sets.[recall that \(A \in \mathscr{A}\) is an
Let \((X, \mathscr{A})\) be a measurable space and \(\left(\mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a strictly increasing sequence of \(\sigma\)-algebras, i.e. \(\mathscr{A}_{n} \subsetneq
Verify the properties \(\left(\mathscr{O}_{1}ight)\)-( \(\left.\mathscr{O}_{3}ight)\) for open sets in \(\mathbb{R}^{n}\). Is \(\mathscr{O}\) a \(\sigma\)-algebra?
Denote by \(B_{r}(x)\) an open ball in \(\mathbb{R}^{n}\) with centre \(x\) and radius \(r\). Show that the Borel sets \(\mathscr{B}\left(\mathbb{R}^{n}ight)\) are generated by all open balls
Let \(\mathscr{O}\) be the collection of open sets (topology) in \(\mathbb{R}^{n}\) and let \(A \subset \mathbb{R}^{n}\) be an arbitrary subset. We can introduce a topology \(\mathscr{O}_{A}\) on
Monotone classes (1). A family \(\mathscr{M} \subset \mathscr{P}(X)\) which contains \(X\) and is stable under countable unions of increasing sets and countable intersections of decreasing sets.is
Let \(X\) be an arbitrary set and \(\mathscr{F} \subset \mathscr{P}(X)\). Show that\[\sigma(\mathscr{F})=\bigcup\{\sigma(\mathscr{C}): \mathscr{C} \subset \mathscr{F} \text { countable sub-family
Extend Proposition 4.3(i)4.3(i), (iv) and (v) to finitely many sets A1,A2,…,AN∈AA1,A2,…,AN∈A.Data from proposition 4.3 Let (X, A,) be a measure space and A, B, An, BnA, nN. Then (1) ANB=0
Is the set function \(\gamma\) of Example 4.5 (ii) still a measure on the measurable space \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\) ? Is it still a measure on the measurable space \((\mathbb{Q},
Let \(X=\mathbb{R}\). For which \(\sigma\)-algebras are the following set functions measures:(i) \(\quad \mu(A)= \begin{cases}0, & \text { if } A=\emptyset \\ 1, & \text { if } A eq
Find an example showing that the finiteness condition in Proposition 4.3 (vii) or Lemma 4.9 is essential.[ use Lebesgue measure or the counting measure on infinite tails \([k, \infty) \downarrow
Find a measure \(\mu\) on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\) which is \(\sigma\)-finite but assigns to every interval \([a, b)\) with \(b-a>2\) finite mass.
Let \((X, \mathscr{A})\) be a measurable space and assume that \(\mu: \mathscr{A} ightarrow[0, \infty]\) finitely additive and \(\sigma\)-subadditive. Show that \(\mu\) is \(\sigma\)-additive.
Let \((X, \mathscr{A}, \mu)\) be a measure space and \(F \in \mathscr{A}\). Show that \(\mathscr{A} i A \mapsto \mu(A \cap F)\) defines a measure.
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) a sequence of sets with \(\mathbb{P}\left(A_{n}ight)=1\) for all \(n
Null sets. Let \((X, \mathscr{A}, \mu)\) be a measure space. A set \(N \in \mathscr{A}\) is called a null set or \(\mu\)-null set if \(\mu(N)=0\). We write \(\mathscr{N}_{\mu}\) for the family of all
Let \(\lambda\) be one-dimensional Lebesgue measure.(i). Show that for all \(x \in \mathbb{R}\) the set \(\{x\}\) is a Borel set with \(\lambda\{x\}=0\).[ consider the intervals \([x-1 / k, x+1 / k),
Determine all null sets of the measure \(\delta_{a}+\delta_{b}, a, b \in \mathbb{R}\), on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\).
Verify the claims made in Remark 5.2Data from remark 5.2 As for o-algebras, see Properties 3.2, one sees that 0 and that finite disjoint unions are again in 2: D, E = D, DnE=0 DUE 9. Of course,
The following exercise shows that Dynkin systems and \(\sigma\)-algebras are, in general, different. Let \(X=\{1,2,3, \ldots, 4 k-1,4 k\}\) for some \(k \in \mathbb{N}\). Then \(\mathscr{D}=\{A
Let \(\mathscr{D}\) be a Dynkin system. Show that for all \(A, B \in \mathscr{D}\) with \(A \subset B\) the difference \(B \backslash A \in \mathscr{D}\). [ use \(R \backslash Q=\left((R \cap Q)
Let \(\mathscr{A}\) be a \(\sigma\)-algebra, \(\mathscr{D}\) be a Dynkin system and \(\mathscr{G} \subset \mathscr{H} \subset \mathscr{P}(X)\) two collections of subsets of \(X\). Show that(i)
Let \(A, B \subset X\). Compare \(\delta(\{A, B\})\) and \(\sigma(\{A, B\})\). When are they equal?
Show that Theorem 5.7 is still valid, if \(\left(G_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{G}\) is not an increasing sequence but any countable family of sets such that\[\text { (1) } \bigcup_{n
Show that the half-open intervals \(\mathscr{J}\) in \(\mathbb{R}^{n}\) are stable under finite intersections. [ check that \(\left.\mathrm{X}_{i=1}^{n}\left[a_{i}, b_{i}ight) \cap
Dilations. Mimic the proof of Theorem 5.8 (i) and show that \(t \cdot B:=\{t b: b \in B\}\) is a Borel set for all \(B \in \mathscr{B}\left(\mathbb{R}^{n}ight)\) and \(t>0\).
Invariant measures. Let \((X, \mathscr{A}, \mu)\) be a finite measure space where \(\mathscr{A}=\sigma(\mathscr{G})\) for some \(\cap\)-stable generator \(\mathscr{G}\). Assume that \(\theta: X
Independence (1). Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\mathscr{B}, \mathscr{C} \subset \mathscr{A}\) be two sub- \(\sigma\) algebras of \(\mathscr{A}\). We call
Approximation of \(\sigma\)-algebras. Let \(\mathscr{G}\) be a Boolean algebra in \(X\), i.e. a family of sets such that \(X \in \mathscr{G}\) and \(\mathscr{G}\) is stable under the formation of
Let \(\mu^{*}\) be an outer measure on \(X\), and let \(A_{1}, A_{2}, \ldots\) be a sequence of mutually disjoint \(\mu^{*}\)-measurable sets, i.e. \(A_{i} \in \mathcal{A}^{*}, i \in \mathbb{N}\).
Consider on \(\mathbb{R}\) the family \(\Sigma\) of all Borel sets which are symmetric w.r.t. the origin. Show that \(\Sigma\) is a \(\sigma\)-algebra. Is it possible to extend a pre-measure \(\mu\)
Completion (2). Recall from Problem 9.14 that a measure space \((X, \mathscr{A}, \mu)\) is complete if every subset of a \(\mu\)-null set is a \(\mu\)-null set (thus, in particular, measurable). Let
The steps below show that the family \(\epsilon_{\lambda}(t):=e^{-\lambda t}, \lambda, t>0\), is determining for \(([0, \infty), \mathscr{B}[0, \infty))\).(i) Weierstraß' approximation theorem
Show that we even get ' \(=\) ' in the estimate denoted by (ii) in Lemma 18.9.Data from lemma 18.9 Lemma 18.9 Let (X, d) be a metric space, the open sets, the closed sets and P = P(X) the power set.
Let \(X=\mathbb{R}^{n}\) (or a separable metric space). Let \(B\) be a Borel set or, more generally, an \(\overline{\mathcal{H}}^{\phi}\)-measurable set, such that \(\mathcal{H}(B)[Instructions. Open

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