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

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Product measures and absolute continuity

For $j=1,2$, let $\mu_j,\nu_j$ be $\sigma-$finite measures on $(X_j, M_j)$ such that $\nu_j<<\mu_j$. Prove that $\nu_1 \times \nu_2 << \mu_1 \times \mu_2$. Here, does it suffices to prove ...
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Reference requests for an opt-cited result in Jennrich (1969)

Lemma 2 on page 637 of Jennrich (1967) states that: Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. ...
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Challenge on one Exam in Real Analysis [on hold]

Could anyone help me in the following midterm exam? Suppose $\{A_n \}\subseteq \mathscr{A}$ and $\cdots\subseteq\,A_{n+1} \subseteq A_n \subseteq \ldots \subseteq A_1 $. Show that ...
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Inner Measure: Measurability

Reference This problem grew out from: Inner Measure vs. Outer Measure Setting Given a plain space $\Omega$ and a possibly empty semiring $\mathcal{S}$. Consider a premeasure ...
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Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; ...
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Image, preimage and set operation in mappings

Not the best title but I don't know how to better describe it. So the image of a set is usually written as $f(B)$, my question is, can I use sets in the place of variables in the expression of my ...
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1answer
18 views

example of a sequence $f_n$, $n=1,2…$ of integrable functions converging to $f$ s.t limit of integral of $f_n$ does not exist

Is there an example of a sequence of of functions $f_n$ converging to a function $f$ such that $f_n$, $n=1,2...$ are integrable and nonegative and their integral over a measurable set $A$ is less than ...
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Measure Theory Terrence Tao Exercise 1.7.19 (vi)

Show that the Lebesgue $\sigma$ algebra on $\textbf{R}^{d+d^{'}}$ is the completion of the product of the Lebesgue $\sigma$ algebras of $\textbf{R}^{d}$ and $\textbf{R}^{d^{'}}$ with respect to the ...
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Terry Tao Measure Theory Exercise 1.7.22

Exercise 1.7.22. The purpose of this exercise is to demonstrate that Tonelli's theorem can fail if the $\sigma$-finite hypothesis is removed, and also that product measure need not be unique. ...
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Intuition behind Strassen's theorem

Currently I am dissecting a proof of Strassen's theorem, which states the following: Suppose that $(X,d)$ is a separable metric space and that $\alpha,\beta>0$. If $\mathbb{P}$ and ...
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1answer
21 views

Convergence in L^p, Cauchy in L infinity

If $u_n$ is a convergent sequence in $L^p$ with $u_n \to u$, and $u_n$ is convergent is $L^\infty$, is it true that the limit in $L^\infty$ must be $u$? Is it true if $u_n$ are all test functions, ...
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33 views

Fatou Theorem what is lim(inf?

We have the following theorem let fn converging to f almost everywhere on measurable A fn, n=1,2.... are integrable and nonnegative and integral of all fn less equal to K.Then f is integrable on A and ...
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1answer
29 views

If $P(A_n) \ge \epsilon>0$ for large $n$, then $P(A_n i.o.) \ge \epsilon$

If $P(A_n) \ge \epsilon>0$ for large $n$, then $P(A_n i.o.) \ge \epsilon$. I tried mimicking the proof of B-C but it give the wrong inequity in a different direction.
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Every $\sigma-$finite measure is semifinite. $(X, \mathcal{M}, \mu)$ is a measure space.

Definition 1: Say $X = \bigcup_{n=1}^{\infty} E_n $ where $E_n \in \mathcal{M}$ and $\mu( E_n ) < \infty $ for all $n$, we call $\mu$ $\sigma$-finite. More generally, if $E = \bigcup^{\infty} E_n ...
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Product Measures Counter Example [duplicate]

I am aware that the product of two Lebesgue $\sigma$ algebra is not a Lebesgue $\sigma$ algebra. Could someone illustrate this with an example.
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1answer
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Premeasures: Inner Measure vs. Outer Measure

Problem Given a plain space $\Omega$ and a ring $\mathcal{R}$. (In fact, a semiring would do the job, too.) Consider a premeasure $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. For simplicity, ...
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Measurability of $f:X\times Y\to\mathbb{K}$ and $f(-,y):X\to\mathbb{K}$

Let $(X,\mu_x)$ and $(Y,\mu_y)$ be two measure spaces endowed with $\sigma$-additive compete measures $\mu_x$ and $\mu_y$, respectively. Let $\mu:=\mu_x\otimes\mu_y$ be the Lebesgue extension of ...
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1answer
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$ \int_{\mathbb{R}^n} f^p dx$ for $p>0$ and measurable $f$

Let $f: \mathbb{R}^n \rightarrow \mathbb{\overline{R}} $ be non-negative and (Borel)-measurable and $p>0$. Then: $$ \int_{\mathbb{R}^n} f^p dx = p \int_{0}^{\infty} t^{p-1} ...
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34 views

Find a Borel Measure on $\mathbb R$ such that integral over positive measureable function equals $0$ implies the function is $0$.

this is a homework problem and I can't figure it out. Find a Borel measure $\mu$ on $\mathbb R$ such that for all $f \in L^+(\mathbb R)$, $\int_\mathbb R f \, d\mu = 0 \; \implies \; f = 0$ All ...
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Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
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Lebesgue outer measure coincides with another measure

I have this problem For $\varepsilon>0$, let $G_\varepsilon$ be the class of all open oriented rectangles in $\mathbb{R}^n$ with diameter $<\varepsilon$ and let $\tau_\varepsilon(I)=$ volume ...
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Let $\mathcal H$ be a Hilbert space. If $\mathcal H$ is not finite-dimensional, then $B := \{x \in \mathcal H : ||x|| \le 1\}$ is not compact.

Let $\mathcal H$ be a Hilbert space. Consider $B := \{x \in \mathcal H : ||x|| \le 1\}$. I've proven that $B$ is closed and bounded with respect to the metric $\rho(x,y) = || x -y ||$. Now I want to ...
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Show $u\in H^1(B(0;1/2))$ is holder continuous, where $u$ is a weak solution to $-\Delta u+cu=f$ for some $c\in L^q$ for some $3/2<q<2,$.

If $u\in H^1(B)$, $B=\lbrace x\in\mathbb{R}^3, |x|<1/2\rbrace$ is a weak solution to $$-\Delta u+cu=f$$ for some $c\in L^q$ for some $3/2<q<2,$ and $f\in C^\infty$, then show $u$ is holder ...
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Measure Theory Problems [duplicate]

Let $\mathcal{E}$ be an arbitrary collection of subsets of a set $X$, let $A$ be a nonempty subset of $X$, and let $$ \mathcal{E}\cap A:=\{E\cap A:E\in \mathcal{E}\}. $$ Show that the $\sigma$-algebra ...
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1answer
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Pre-measure restricted to a set in $\mathcal{A}$.

Definition from Folland-Real Analysis, A function $\mu_0: \mathcal{A} \to [0, \infty]$ will be called a $\textbf{premeasure}$, if (a). $\mu_0(\emptyset)=0$ (b). if $\{A_j\}_1^{\infty}$ is a ...
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1answer
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Confusions about Radon-Nikodym derivative and dominating measures

I have some difficulties to understand the Radon-Nikodym derivative and link it to the ordinary way of obtaining the probability density function, which is through the derivative of cumulative ...
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How can we go from uniform integrable to uniform L1

To be specific, if we have a set of $\{f_n\}$ measurable functions, uniform integrable: For any $\epsilon>0$, we can find a $\delta$, such that for any set E with $\mu\{E\}<\delta$, $|\int ...
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1answer
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Does the statement “$f = 0$ almost everywhere” depend on the measure that is defined?

I know the convention is to use the Lebesgue measure but is there ever a situation where we would interpret "$f(x) = 0$ almost everywhere" by using a different measure? For example, let $f(x) = 1$. ...
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1answer
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Limit of an Indicator function

According to an instructor, the following is true: $$\lim_{n \to \infty} \mathbf{1_{\mathrm{[n,n+1]}}} = \mathbf{1_{[\varnothing]}} =0,$$ where $\mathbf{1}$ is the indicator function and the limit is ...
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Product Measure in $\mathbb{L}^p$

Let $(X,\mathscr{A},\mu_1)$ and $(Y,\mathscr{B},\mu_2)$ be $\sigma$-finite measure spaces and let $\nu$ be the product measure. Let $p\in[1,\infty)$, $f\in L^p(\mu_1), g\in L^p(\mu_2)$, and define $(f ...
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Structure of the space $X:=\{x \subset \mathbb{R}^3: |x| < \infty\}$, where $|x|$ is the cardinality of $x$

I am interested in the space $$ X:=\{x \subset \mathbb{R}^3: |x| < \infty\}, $$ where $|x|$ is the cardinality of the subset $x$. This is basically configuration space for a quantum system with a ...
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2answers
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Generators of the Borel $\sigma$-algebra on $\mathbb{R}^2$

How do I show that a set of closed sets (plus the empty set) is a generator for $\mathbb{B}_2$? The set in question is the set made of set of vectors in a given range of angles and lengths, think of ...
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1answer
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Monotone Class Theorem Application

I am trying to proof the following statement. Let $h$ be a bounded, $\mathbb{F}$-predictable process with $\tau$ a $\mathbb{H}$-stopping time, we then like to prove \begin{equation} ...
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2answers
28 views

Is every Lebesgue measurable function bounded on a set of positive measure

Let $f$ be a Lebesgue measurable function from $[0,1]\to\mathbb{R}$. Let $\mu$ be Lebesgue measure. Does there exist a measurable set $B$ with $\mu(B)>0$ and an $M>0$ such that for all $x\in B$, ...
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1answer
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Constructing an outer measure on a collection of subsets

Let $X$ be the set of three elements $\{a,b,c\}$ . On the collection of subsets $C = \{\{ \emptyset \} , \{a\} , \{a,b\}\}$ Define the set function $m: C → [0,∞]$ by $m( \emptyset ) = 0 $, ...
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Finite measure and a measure $>0$?

This is a problem from one of the analysis qualifying exams in my school Let $\mu$ be a $\sigma$-finite measure on $(X,F)$ with $\mu(X) = \infty$. Show that for every $C > 0$, there exists an $E ...
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1answer
38 views

Lebesgue point and integration

Let $f$ be in $L^1_{\text{loc}}(\mathbb{R})$. We know that for almost every $t$ $$ \lim_{h\to 0} \frac{1}{h} \int_t^{t+h} |f(u)-f(t)|\text{d} u = 0. $$ My question is : can we say that for almost ...
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1answer
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Development of Measure Theory

I would like to see the historical references for the following sequence of events: 1) When outer measure defined first time? 2) When it is proved that the outer measure is not countable additive? ...
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Cauchy's Functional Equation

Consider Cauchy's Functional Equation $$\phi(t+s)=\phi(t)+\phi(s).$$ Can we say that any right continuous with left limits (cadlag) solution is Borel measurable? Obviously continuous solutions are ...
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2answers
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Existence of strictly positive probability measures

Let $X$ be a Hausdorff space (or let's even assume it is metrizable). A strictly positive measure on $X$ is the that gives positive measure to any non-empty open subset of $X$. Under which condition ...
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Measurable set contains a sequence

I found this question and didn't manage to extrapolate from the hint, could anyone help? Here's the question for the sake of completeness: Let $A\subseteq[a,b]$ be Lebesgue measurable, such that: ...
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Showing that $\lim_n \sum_{k = 0}^n a_{kn} \lambda_1(A_{kn}) = \int_a^b f d\lambda$.

I need some help with the following: (i) Show that $\chi_{\mathbb Q \cap [a,b]}$ is measurable with respect to $\mathcal B(\mathbb R)|_{[a,b]}$ and compute $\int \chi_{\mathbb Q \cap [a,b]} ...
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2answers
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Measure-theoretic analog of homeomorphism and isometry

If $(X,\tau_X)$ and $(Y,\tau_Y)$ are topological spaces and $f:X\to Y$ is a continuous bijective function between them such that $f^{-1}$ is also continuous, then the two topological spaces are said ...
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1answer
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Equivalency in the elementary measure theory

Show that: $f\geqslant0$ and $\int f =0 $ $\Leftrightarrow$ $\mu$({$x$$\in$$X:$ $f($x$)>0$})=$0$ My idea: Let {$x$$\in$$X:$ $f($x$)>n$}=$E_{n}$ $\mu$({$x$$\in$$X:$ ...
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Can the integral get small outside a set with finite measure?

Let $(X, \mathcal{A}, \mu )$ be a measure space and let $ f : X \rightarrow \mathbb{\overline{R}}$ integrable. I just proved the fact that for every $\epsilon > 0$ we find $\delta > 0$ so that ...
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Continuity from below?

Let $(\mathbb{R}^2,\mathcal{B}(\mathbb{R}^2))$ be our space. I have the set of "arc segments", $A(\theta,\eta,r,R)$, where $0\le\theta\le\eta<2\pi, \text{and } 0<r\le R$. The elements in the ...
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how that T is ergodic if and only if the only eigenfunctions $f \in L^2(\mu)$ of $U_T$ corresponding to the eigenvalue $1$ are constant functions.

Let $T:X \rightarrow X$ be a measure-preserving transformation. Assume that $(X,\mathcal{B},\mu)$ is a probability space. Show that T is ergodic if and only if the only eigenfunctions $f \in ...
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Borel set's incompleteness

I was told that while M, class of all Lebesque measurable set is complete, sigma field generated by all open set B is not. Can someone give me an example of interval which is inside M yet is not ...
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1answer
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Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
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1answer
29 views

Showing that Lebesgue measure is preserved by translations of the $d$-dimensional torus

Let $\underline{\alpha}=(\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$. Show that the transformation $R_{\underline{\alpha}}=\mathbb{T}^d \rightarrow \mathbb{T}^d$ defined by ...