Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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1answer
131 views

Lebesgue measure on $\mathbb{R}/\mathbb{Z}$

I was reading a (brief) introduction about measure theory today and came across the following statement: (Lebesgue measure on $\mathbb{R}/\mathbb{Z}$): There is a unique probability measure $\mu$ ...
2
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2answers
87 views

Question about $L^p$ spaces

Suppose $1<p<\infty$ and let $L^1$ and $L^p$ denote the usual Lebesgue spaces on $[0,1]$. Let $$A=\{f\in L^1:\|f\|_p\leq 1\}.$$ Show $A$ is closed in $L^1$. I took a sequence $\{f_n\}$ ...
2
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0answers
201 views

$p$-norm is a convex function of $p$

For what measures $\mu$ and what intervals $(a,b) \subset (1,\infty)$ is the function $$p\mapsto \|f\|_p =\left(\int |f|^p d\mu \right)^{2/p}$$ a convex function of $p$ on $(a,b)$ for all $f\in ...
4
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1answer
297 views

Weak*-convergence of regular measures

Let $K$ be a compact Hausdorff space. Denote by $ca_r(K)$ the set of all countably additive, signed Borel measures which are regular and of bounded variation. Let $(\mu_n)_{n\in\mathbb{N}}\subset ...
6
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1answer
513 views

stopped filtration = filtration generated by stopped process?

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
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2answers
75 views

Measure spaces such that the semi-norm is a norm

In my lecture notes there is the following exercise: "Characterize those measure spaces $(X, B, \mu)$ on which the semi-norm $\|f\| = \int_X |f| d \mu$ defined on $L^1(X) = \{ f \mid f \text{ ...
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1answer
101 views

How to show that $\int \phi \,d\mu=-\int\phi'(x)f(x)\,dx$

Assume that $f\colon \Bbb R \rightarrow\Bbb R$ is left-continuous nondecreasing and let $\mu$ be a Borel measure in $\Bbb R$ such that $\mu([a,b))=f(b)-f(a)$ for $a<b$, $a,b \in\Bbb R$. I would ...
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3answers
1k views

Could someone remind me why is incorrect to switch an infinite sum and an integral?

Could someone jog my memory on this? The order of operation between an $\int$ and $\sum_{n\in \mathbb{N}}$ is not always interchangable? Note that the sum is an INFINITE sum Why is it that $\int ...
8
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1answer
525 views

Inclusion of $L^p$ spaces

Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some ...
4
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1answer
367 views

Infinite-dimensional translation-invariant measure

Why is there no translation-invariant measure on an infinite-dimensional Euclidean space? Is there a reasonably short, insightful proof? I am interested in an infinite-dimensional space with a ...
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1answer
424 views

interchange sum and integral

suppose I have a family of i.i.d standard normal random variables $Y_{n,k}$ and I define $X^N_t:=\sum_{n=0}^N\sum_{k=1}^{2^n}Y_{n,k}\phi_{n,k}(t)$ for $t\in [0,1]$ where $\phi_{n,k}$ are the Schauder ...
1
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1answer
128 views

Explanation of statement needed (Bochner-style integral, Fubini's theorem, etc.)

I am reading a paper. They define $$ L_{p,q}(Q) = \{ u \in L_p((0,T); L_q(Y)) : u(t, \cdot) = 0 \text{ on } Y \backslash Y_t \text{ for a.e. $t \in (0,T)$}\}$$ with norm $$\lVert u ...
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0answers
111 views

Completeness of $L^1$

I have written a proof that $L^1$ is complete. Can you read it and tell me if it's right? Thanks. To show $L^1$ is complete we use the following fact: Fact: If $f_n$ is a sequence in $L^1$ such that ...
2
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1answer
48 views

Give a example about invariant ergodic measure and quasi-symmetric mapping

Is there a example $(X,f,\mu)$ such that $X$ is a closed subset of Euclidean space, $f$ be a quasi-symmetric mapping but not a Lipschitz mapping, $f(X)=X$, $\mu$ is a finite measure on $X$ that is ...
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2answers
1k views

set in $\mathbb{R}$ which is not a Borel-set [duplicate]

Possible Duplicate: Lebesgue measurable but not Borel measurable Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly if i start from the topology of $\mathbb{R}$, i.e. all ...
2
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1answer
96 views

When does non-negativity of the integral of a function imply that the function itself is non-negative?

Let $(\Omega,\Sigma)$ be a measurable space and $(\omega_k)_{k\in\mathbb{N}}$ a sequence of elements of $\Omega$. Let $$ \mathcal{M}:=\left\{\sum_{k=1}^\infty a_k\cdot\delta_{\omega_k}: ...
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1answer
79 views

Question about an implication in a theorem

There is the following theorem: If $(f_n)$ is a sequence in $L^1$ such that $\sum \|f_n\|_1 < \infty$ then (1) $\sum f_n $ converges almost everywhere (i.e. $\sum f_n(x) = K_x < \infty $) ...
2
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2answers
136 views

Wrong proof of convergence almost everywhere

Can you tell me where the mistake is? If $(f_n) \in L^1$ is a sequence of functions such that $\sum_n \|f_n\|_1 < \infty$ I can prove that $f(x) = \sum_{n=1}^\infty f_n(x) < \infty$ for all $x ...
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2answers
414 views

Sequence of Uniformly Bounded functions

Consider a sequence $\{ f_k \}_{k=1}^{\infty}$ of locally-bounded functions $f_k: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$. Assume the following. For any sequence $\{X_k\}_{k=1}^{\infty}$ of ...
4
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2answers
409 views

Measurable homomorphism of $\mathbb{T}$ into $\mathbb{C}^\times$

Working through Katznelson's An Introduction to Harmonic Analysis and have been stumped by the following problem for the past few days: Show that a measurable homomorphism of ...
2
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3answers
775 views

Give an example of a measure which is not complete

Give an example of a measure which is not complete ? A measure is complete if its domain contains the null sets.
2
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2answers
83 views

Prove $\mu(\{|f-\mu f|>K\}) \le \frac{1}{K^2}(\mu f^2 -(\mu f)^2)$

Let $f$ be a random variable on a probability space $(\Omega, \Sigma,\mu)$ where $\mu f^2 < \infty$. How would I prove (or disprove) that $$\mu(\{|f-\mu f|>K\}) \le \frac{1}{K^2}(\mu f^2 -(\mu ...
4
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2answers
160 views

Uniqueness of extension of zero measure

Let $(\Omega,\mathscr F)$ be a measurable space with two probability measures $\mu, \nu: \mathscr F\to[0,1]$ defined over it. Suppose that $\mathscr C\subset\mathscr F$ is some class of sets and $$ ...
0
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1answer
469 views

What is the minimum $ \sigma$-algebra that contains open intervals with rational endpoints

What are the minimum $\sigma$-ring and $\sigma$-algebra on $\mathbb R$ which contain the open intervals with rational endpoints? Is there a relation between this $\sigma$-algebra and Borels?
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3answers
108 views

sequence of open intervals

Let $ \displaystyle{ \{ (a_n , b_n) : n \in \mathbb N \} }$ a sequence of open intervals on $\mathbb R$ such that $ \displaystyle{[0,15] \subset \bigcup_{n=1}^{n_0} (a_n ,b_n) }$ for some $ n_0 \in ...
1
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3answers
108 views

lebesgue measure of $\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, 0 \leq y \leq 10, z \in \mathbb Z \} $

Find the lebesgue measure of the set: $$ \Bigl\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, \quad 0 \leq y \leq 10, \quad z \in \mathbb Z \Bigr\} $$ I think is a null set but for some reason I ...
3
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1answer
90 views

If $f\in C^1, L^1$, does $\lim_{x\rightarrow \infty} f(x)=0$?

Problem: Prove or provide a counterexample: Let $m$ be the Lebesgue measure on $\mathbb{R}$. Suppose $f\in L^1(\mathbb{R}, m)$ is of class $C^1$, and that $f'\in L^1 (\mathbb{R}, m)$. Then ...
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1answer
140 views

Conditional expectation independence

I'm working on some statistics project and am not getting further because of some stupid prediction that doesn't want to be 0. That's why I was wondering if maybe the following holds: Suppose we have ...
5
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1answer
1k views

Set of convergence is measurable. [duplicate]

Possible Duplicate: pointwise convergence in $\sigma$-algebra Problem: Prove that the set of points at which a sequence of measurable real functions converges is a measurable set. (I ...
0
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1answer
114 views

Prove that a product of nullsets in $\mathbb{R}^n$ by a compact set in $\mathbb{R}$ is a nullset in $\mathbb{R}^{n+1}$

Let $K$ be a compact set, $K \subset \mathbb{R}^n \times [a,b]$ and, for each $t \in [a,b]$ define $K_t = \{ x\in \mathbb{R}^n $ ; $(x,t) \in K\}$. If $\forall t \ K_t$ has measure zero in ...
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1answer
203 views

Absolute continuity between measures, one induced from the other by a measurable mapping?

$(\Omega, \mathcal{F},\mu)$ is a measure space, and $f: \Omega \to \Omega$ is a measurable mapping. Let $\nu$ be the measure on the same measurable space induced from $\mu$ by $f$ . I wonder if there ...
0
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1answer
203 views

Composition of Measurable and Discontinuous function

Consider a locally-bounded function $f: W \rightarrow X$, $X \subseteq \mathbb{R}^n$, $W \subseteq \mathbb{R}^m$. Assume that $f$ is Borel measurable (for every open $O \in \Sigma_X$ ...
4
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1answer
509 views

Problem from Brezis's book (mollifiers)

Any ideas on how to get started with this? Let $\rho \in L^1(\mathbb{R}^N)$ with $\int \rho=1$. Set $\rho_n(x)=n^N \rho(nx)$. Let $f \in L^p(\mathbb{R}^N)$. Show that $\rho_n \star f \to f$ in ...
2
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1answer
372 views

Show there exists a positive, strictly increasing measurable function

Let $f$ be a nonnegative measurable function on $[0,\infty)$ such that $\int_0^\infty f(x)dx < \infty$ is finite. Show that there is a positive, strictly increasing measurable function $a(x)$ on ...
2
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1answer
328 views

Composition of measurable functions

Consider a locally bounded function $f: X \times W \rightarrow X$, where $X \subseteq \mathbb{R}^n$, $W \subseteq \mathbb{R}^m$, such that for all $x \in X$ the function $w \mapsto f(x,w)$ is (Borel) ...
1
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1answer
569 views

bounded measurable function is the uniform limit of a sequence of simple functions

Let $ f: \mathbb R \to \mathbb R $ a non-negative bounded measurable function. Prove that there exists a sequence of simple non-negative functions $ (f_n)_{n \in \mathbb N} $ such that $ f_n \to f$ ...
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1answer
255 views

Measurable subset of unit circle invariant under translation by infinitely many points.

I'm working through Katznelson's An Introduction to Harmonic Analysis. Currently, I'm looking at an exercise in the first chapter: Prove that if $E$ is a measurable set on ...
0
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1answer
243 views

Measure of “boundary” of a nowhere dense set

Suppose $E$ is a nowhere dense set. For simplicity, assume it is in $R$. Is it true that the Lebesgue measure of $\overline{E}-E$ is zero? I.e., $m(\overline{E}-E)=0$. The statement is not true in ...
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1answer
108 views

Radon measures and sequences

I have a question about Radon meaures: Given a Radon measures $ \mu_{1}, \mu_{2}$, both have compact support: How to show that $\int \hat{\mu_{1}}(x)\,d\mu_{2}(x)=\int \hat{\mu_{2}}(x)\,d\mu_{1}(x)$ ...
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1answer
202 views

Induced $\sigma$-algebra vs. product $\sigma$-algebra

Let $(E,\mathscr E)$ be a measurable space and $(S,2^S)$ be a finite set. Let $$ \xi:(E,\mathscr E)\to(S,2^S) $$ be a mesaurable function, i.e. $\xi^{-1}(s)\in \mathscr E$ for any $s\in S$. Now, ...
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0answers
140 views

Under certain condition, a local martingale is a martingale

It's well known that a local martingale of is a uniformly martingale if and only if it is of class D. I want to show the following: Let $L$ be a continuous local martingale, null at zero such that ...
2
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1answer
166 views

Show Fubini is not appliable

I have the following integrals $$\int_0^1 \int_1^\infty e^{-xy} - 2 e^{-2xy} \lambda(dx) \lambda(dy)$$ and $$\int_1^\infty \int_0^1 e^{-xy} - 2 e^{-2xy} \lambda(dy) \lambda(dx)$$ and I shall prove, ...
5
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0answers
224 views

Existence of measures assigning positive values to all open sets

Let $K$ be a compact Hausdorff space. Does there exist a finite Borel measure on $K$, assigning positive values to all non-empty open sets of $K$?
4
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1answer
127 views

How to show $\mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R}) \subset \mathcal{L}(\mathbb{R^2})$?

Let $\mathcal{L}(\mathbb{R})$ be the lebesgue-measurable set of $\mathbb{R}$, and $\mathcal{L}(\mathbb{R^2})$ the lebesgue-measurable set of $\mathbb{R^2}$. First I shall show, that ...
4
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1answer
56 views

Measurability of a map

Let $(X,\mathscr F)$ be some measurable space and $Y$ be a finite set with a $\sigma$-algebra $2^Y$. Let the map $$ f:X\to Y $$ be $\mathscr F|2^Y$-measurable. Consider sets $X^\mathbb N$ and ...
6
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1answer
2k views

weak convergence in $L^p$ plus convergence of norm implies strong convergence

Having trouble with this problem. Any ideas? Let $\Omega$ be a measure space. Let $f_n$ be a sequence in $L^p(\Omega)$ with $1<p<\infty$ and let $f \in L^p(\Omega)$. Suppose that $$f_n ...
4
votes
2answers
148 views

$p \leqslant q \leqslant r$. If $f \in L^p$ and $f \in L^r$ then $ f \in L^q$? [duplicate]

Possible Duplicate: Proving an interpolation inequality Let $f \in L^p$ and $f \in L^r$ where $1 \leqslant p \leqslant r$ . Then can we say that $f \in L^q$ if $p \leqslant q \leqslant r$? ...
1
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1answer
241 views

Convergence in the absence of Dominated Convergence Theorem, and uniform integrability

This question is extended from Resnick's exercise 5.13 in his book A Probability Path. Let the probability space be the Lebesgue interval, that is, $(\Omega=[0,1],\mathcal{B}([0,1]),\lambda)$ and ...
1
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2answers
94 views

Convergence of Expectation for $X_n=\frac nY 1_{\{Y>n\}}$ for any $Y$ such that $P(0\le Y<\infty)=1$

Here is another self-study exercise that I am struggling mightily with: $X_n=\frac nY 1_{\{Y>n\}}$ for any $Y$ such that $P(0\le Y<\infty)=1$ I am told that $X_n\to X$ a.s for some $X$, and am ...
6
votes
2answers
375 views

Why define measures on $\sigma$-rings?

I have the impression that modern texts deal almost excusively with measures on $\sigma$-algebras, while older texts, such as the one of Halmos, deal mainly with measures defined on $\sigma$-rings. ...