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

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On proving that a infinite intersection of truth sets is empty and on the usefulness of almost surely.

I am trying to solve the exercise at the end of this page, the framework is that of measure theory where we are tossing a coin infinitely often so we are working with a probability triple $( \Omega, ...
3
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0answers
73 views

Prove that $\lim\limits_{n \to \infty } \int_0^\infty (1 + x/n)^{-n}x^{-1/n}dx= 1$ using DCT

Prove that $\mathop {\lim }\limits_{n \to \infty } \int_0^\infty {\frac{{dx}}{{{{(1 + \frac{x}{n})}^n}{x^{\frac{1}{n}}}}}} = 1$ using dominated convergence theorem (DCT). By DCT we need to show ...
-4
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2answers
56 views

Prove that if $f^2$ is integrable in $X$, then $f$ is integrable in $X$.

I am studying for a test in measure theory. Please help with the following question: $(X,A,\mu)$ a finite measure space, and $f$ is a measurable function in $X$. Prove that if $f^2$ is integrable in ...
5
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2answers
76 views

A question about measure set

Suppose that a sequence of sets $\{A_n:n\in \Bbb N\}$ is increasing, and $A=\bigcup_{n=1}^\infty A_n$. If $A$ is measurable, $\mu(A)\gt 0$ and $\mu$ is an atomless measure, do there exist an $n\in ...
1
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1answer
30 views

Measurability of $t \mapsto \int_{\Omega(t)}f(t)g(t)h(t)$ given measurability of $t \mapsto \int_{\Omega(t)}f(t)g(t)$?

Suppose I know that, given $f(t), g(t) \in L^2(\Omega(t))$, $$t \mapsto \int_{\Omega(t)}f(t)g(t)$$ is measurable on $([0,T], Lebesgue) \to (\mathbb{R}, Borel)$. Suppose $h(t) \in L^\infty(\Omega(t))$ ...
5
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1answer
39 views

A sufficient condition for almost everywhere equality

Let $f,g:(0,\infty)\to \mathrm{R}$ be monotone decreasing functions. Show that if $m(\{x:f(x)>a\})=m(\{x;g(x)>a\}),\; \forall a\in \mathrm{R}$ where $m$ denotes Lebesgue measure, ...
2
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1answer
29 views

limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
2
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2answers
110 views

Inclusions relating standard norms, in measury theory

I know that for finite measure space $(X, \mathcal A ,\mu )$ and $1\leq p< q<\infty $ , the inclusion $\mathcal L^q\subseteq \mathcal L^p\subseteq\mathcal L^1 $ holds true (applying Holder's ...
2
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1answer
57 views

On the equality of two sets (a doubt from Probability with Martingales).

Let $(S, \Sigma, \mu) $ be $([0,1], \mathcal{B}[0,1], Leb)$. Let $\epsilon(k)$ be a sequence of strictly positive numbers s.t. $\epsilon(k) \downarrow 0$. Let $V = Q \cap [0,1],$ the set of rationals ...
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2answers
48 views

lebesgue measure basic exercise

I have a basic question about a Lebesgue measure exercise that I am not sure how to solve. (I apologize if this is a simple question, I am new with this subject). Compute the Lebesgue measure of $X$ ...
5
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2answers
68 views

for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

This is an old preliminary exam problem: Show that, for every nonnegative Lebesgue integrable function $f:[0,1]\rightarrow \mathbb{R}$ and every $\epsilon>0$ there exists a $\delta>0$ such ...
4
votes
5answers
74 views

Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $). I know about ...
4
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2answers
62 views

A problem on product measure

Let $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$ be two totally finite measure spaces (which implies that $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras). (As usual ...
0
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1answer
35 views

Is the function $\ln (u(x))$ integrable when $u$ is bounded and positive?

Consider $\Omega$ an open bounded domain in $\mathbb{R}^n$ and $ u \in L^{\infty} (\Omega)$ a positive function. My question is : the well defined function $\ln (u(x))$ is integrable? Intuitively ...
8
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2answers
67 views

Lebesgue Measure of Image of Unit Square under Continuous Map

Problem. Let $h\in C(\mathbb{R})$ be a continuous function, and let $\Phi:\Omega:=[0,1]^{2}\rightarrow\mathbb{R}^{2}$ be the map defined by \begin{align*} ...
0
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0answers
39 views

finite signed measure on [0,1]

I'm studying for a qualifying exam and I'm stuck with the following question from an old exam; any help would be greatly appreciated: is there a finite signed measure $\mu$ on $[0,1]$ such that $ \int ...
3
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0answers
55 views

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

Problem Let $f_n\in C[0,1]$. Show that $f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$. Background Let $X$ be a normed space. ...
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2answers
91 views

When is the union of $\sigma$-algebras atomless?

Suppose that we are given a probability space $(\Omega, \mathcal{F}, \mathsf P)$ and an increasing sequence of $$\mathcal{F}_1\subset \ldots \subset\mathcal{F}_n\subset \mathcal{F}_{n+1} \subset ...
3
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2answers
40 views

A weaker form of Lebesgue's differentiation theorem in $\Bbb R ^n$

If $f : \Bbb R ^n \to \Bbb C$ is locally-integrable then Lebesgue's differentiation theorem says that $$\lim \limits _{r \to 0} \frac 1 {\lambda \big( B(x, r) \big)} \int \limits _{B(x, r)} f \Bbb d ...
0
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1answer
35 views

Dunford Pettis Theorem

Suppose $L_1([0,1],\lambda)=L_1(\lambda)$ is the set of all $1$-integrable functions on $[0,1]$. $$S=\{(f_1,f_2)\in L_1^2(\lambda) |0\leq f_1+f_2\leq 1, a.e. \}$$ By Dunford Pettis theorem, we know ...
2
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1answer
41 views

Weak convergence and integrals

Assume $$u_k\rightharpoonup u,\quad v_k\rightharpoonup v\quad\text{in}\quad L^1(0,T;Y)\tag{1}$$ and $$\int_0^T u_k(t)\varphi'(t)\ dt=-\int_0^T v_k(t)\varphi(t)\ dt\tag{2}$$ for some $\varphi\in ...
3
votes
2answers
34 views

A proof regarding Lebesgue measure and a differentiable function

Could anyone kindly provide a hint on the following problem? My guess is to do some change of variables? Thank you! Let $f:\Bbb{R}\rightarrow\Bbb{R}$ be a continuously differentiable function. ...
4
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1answer
40 views

$\pi-\lambda$ Theorem to show measure giving interval lengths equivalent to Lebesgue on [0,1]

so I have been working on this problem and I want to make sure I am understanding the conclusion fully. So I have the following scenario: Not part of the actual question, but relevant. Consider ...
1
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1answer
23 views

Apply Fubini's theorem to measurable equation to switch variables

I would like to understand the following reasoning: Let $m$ be a measurable function on $\mathbb R$ and let E be an equation which is described by $m$ where we insert on boths sides of the equations a ...
1
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0answers
36 views

Krylov-Bogoliubov theorem in discrete and continuous time

Consider the following two similar frameworks (the first one in discrete time, the second one in continuous time) in which I am trying to apply the Krylov-Bogoliubov theorem. 1) (Discrete time.) ...
0
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1answer
21 views

Question about proof that Vitali set is not measurable

I am reading through a proof that the Vitali set is not measurable. I have gotten to a line that says $$[0,1] \subset \cup_{q \in [-1,1] \cap Q} (A+q)$$ and I don't see why this is true. Clearly, ...
7
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1answer
107 views

need reference for fact about products of measures

Need a reference (textbook or paper) for the following (probably well known) fact: Suppose $(X,M)$ is a measurable space and $\lambda$, $\nu$ are two different probability measures on $(X,M)$. Write ...
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1answer
17 views

Supremum over separable Banach space of a measurable function is measurable

Let $X$ be a separable Banach space. Suppose that $f:[0,T] \times X \to \mathbb{R}$ is such that $t \mapsto f(t,x)$ is measurable. Is the function $$t \mapsto \sup_{x \in X}f(t,x)$$ also measurable? ...
0
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1answer
39 views

Equivalence of two definitions of Measurable Sets

(1) A subset $A$ of $\mathbb{R}^d$ is measurable if given $\epsilon>0$, there exists an open set $\mathcal{O}$, such that $A\subseteq \mathcal{O}$ and $m^*(\mathcal{O}-A)<\epsilon$. (2) A ...
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2answers
153 views

Prove that if $B = \{x-y : x,y \in A\}$, where $A$ is a Borel measurable subset of $R$ with positive measure

Suppose that $m$ is Lebesgue measure, and $A$ is a Borel measurable subset of $R$ with $m(A) > 0$. Prove that if $B = \{x - y : x,y \in A\}$, then $B$ contains a non-empty open interval centered at ...
1
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1answer
37 views

Motivic measure

Somebody can give me some good references for start to read Motivic-measure, Now I`m studing the Grothendieck Ring, and is necesary undertand something of motivic theory for my case, so I need a good ...
2
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2answers
32 views

Are there two different notions of “conditional probability”?

This question comes from reading the discussion here. (1) If one is given a "probability measure" $P : F \rightarrow [0,1]$ mapping a Borel $\sigma$-algebra $F$ to $[0,1]$ then for two ``random ...
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1answer
24 views

Control of $L^{\infty}$ Norm of 3d Heat Equation Solution for $L^{3}$ Initial Data

Let $w_{t}$ denote the 3-dimensional heat kernel $$w_{t}(x)=(4\pi t)^{-3/2}e^{-\left|y\right|^{2}/(4t)},\qquad y\in\mathbb{R}^{3}, \ t > 0$$ Suppose $f\in L^{3}(\mathbb{R}^{3})$, and let ...
2
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2answers
59 views

I am having trouble solving this problem from the book “ Measure Theory” by Donald L.Cohn.

Let $(X,\mathcal A ,\mu)$ be a measure space and let $f$ and $f_1 ,f_2 ,....$ be non-negative functions that belong to $\mathcal L^1(X,\mathcal A,\mu,\mathbb R)$ and satisfy- (i) $\{f_n\}$ converges ...
0
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1answer
24 views

Is the class of negligible sets a monotone class?

Let $(X,\Sigma,\mu)$ be a measure space we say that the subset $F$ of a set $X$ is negligible set if there exist $G \in\Sigma$ , $F$ is a subset of $G$ And $\mu(G)=0$ My question is: Is the class of ...
3
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1answer
51 views

When does the diagonal have zero measure?

Let $\mu$ be a Radon measure on a space $X$ and consider the product measure $\mu \otimes \mu$ on $X \times X$. Is there some necessary and/or sufficient condition on $\mu$ to guarantee that the ...
1
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1answer
63 views

Proof that random variable is almost surely constant

If a random variable $X : \Omega \to \mathbb R$ is $\{ \emptyset, \Omega \}$-measurable, then it is constant. I want to generalise this result: Now if $\mathcal G$ is a $\sigma$-algebra such that ...
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1answer
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Problem about integration

Let $\mathcal R$ be a $\sigma$-algebra in a nonempty set $X$, let $\mu$ be a positive measure on $\mathcal R$, let $f:X\to \mathbb C$ be measurable relative to $\mathcal R$,and $f\in L^1(\mu)$. Let ...
3
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1answer
98 views

Does this density limit exist?

Suppose that $T:\mathbb{R}^k\rightarrow\mathbb{R}^k$ is a transformation which is differentiable at a point $x\in\mathbb{R}^k$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^k$. Denote by ...
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0answers
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If $\partial E$ has Jordan outer measure zero, then $E$ is measurable.

I am going through Tao's measure theory book, and have to prove If $\partial E$ has Jordan outer measure zero, then $E$ is measurable. where $\partial E$ denotes the boundary of the set $E$. I ...
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0answers
41 views

Does this limit of measures exist?

Suppose that $T:\mathbb{R}^k\rightarrow\mathbb{R}^k$ is a differentiable transformation. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^k$. Denote by $B(x,r)$ the open ball centred at $x$ with ...
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2answers
29 views

Expectation of Nonnegative Random Variable - Measurability

Recall the result that for a nonnegative random variable $X$ on $(\Omega, \mathcal{F}, P)$, $$ E[X] = \int_0^\infty (1 - F(x)) dx, $$ where $F$ is the cdf of $X$. In many of the proofs I've seen for ...
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1answer
19 views

Are these two statements involving null sets and $L^2$ Bochner functions equivalent?

Suppose I have two functions $f, g \in L^2(0,T;L^2(\Omega))$ where we have some bounded domain $\Omega$. Suppose that $$\text{for almost all $t$,}\quad f(t) \leq g(t) \quad\text{almost everywhere in ...
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0answers
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If the right side of $\int f\ d\lambda = \int f\ d\mu − \int f\ d\nu$ exists, does the left one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
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2answers
37 views

Prove that $X,Y$ are independent iff the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $X$ and $Y$ be random variables on $(\Omega,\mathcal A,\operatorname P)$ with values in $\mathbb{R}^m$ and $\mathbb{R}^n$, ...
2
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1answer
61 views

Is there measurable function defined on unmeasurable set?

In my textbook, Lebesgue measurable function is defined as for every finite $a$, the set $\{x\in E:f(x)>a\}$ is a measurable set of $R^n$. And it further states $E=\{x\in ...
1
vote
1answer
11 views

Characterization of the space of integrable functions stable under multiplication

Let $Ω = (Ω,Σ_Ω,μ)$ be a measure space and let $L(Ω)$ be the space of integrable functions on $Ω$. For $f ∈ L(Ω)$, set $L_f(Ω) = \{φ ∈ L(Ω);~f·φ ∈ L(Ω)\}$. Has the space of all integrable functions ...
1
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2answers
29 views

If $B$ is a BM and $\mathcal F_t=\sigma(B_s,s\le t)$, then $(B_{s+t}-B_t)_{s\ge 0}$ is independent of $\mathcal F_t^+:=\bigcap_{s>t}\mathcal F_s$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
0
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1answer
32 views

$\sigma(f_i, i \in I)$ ($f_i:\Omega \to \mathbb{R}$) where $I$ is uncountable

$\sigma(f_i, i \in I)$ ($f_i:\Omega \to \mathbb{R}$) where $I$ is uncountable contains only sets that can be written as $\{(f_{i_1},f_{i_2},...) \in B\}$ where $B \in ...
0
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1answer
27 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...