Questions tagged [measure-theory]

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

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Cantor Ternary Set Problem, Ternary Expansions

I'm reading Lebesgue Integration on Euclidean Space by Frank Jones and I'm stuck on this problem, of page 41, problem 15. Let $C$ the Cantor Ternary Set. Let $G_{1}=(\frac{1}{3},\frac{2}{3})$, $G_{2}...
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16 views

Discrete measure and Lebesgue measurability

According to wikipedia https://en.wikipedia.org/wiki/Discrete_measure a driscrite measure is defined in the following way: Let's consider a real line $\mathbb{R}$. For some (possibly finite) ...
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$\lambda_f(a)$ (Question 6.40 from Folland's Real Analysis Book)

In his book he defined: $$ \lambda_f(\alpha)=\mu\left(\left\{x\middle||f(x)|>\alpha\right\}\right)$$ And with that in mind, he later defined in the excercise for some $\mu$-measureable $f$ on $X$: $...
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Radon-Nikodym derivative for discrete measures

Could you, please, check the following: Let's consider a real line $\mathbb{R}$ and let's define two measures on it. For some (possibly finite) sequences $s_{1}, s_{2}, \dots$ and $a_{1}, a_{2}, \...
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Measure of union of diagonals

Let $\mu$ be a measure on $[0,1]$ which induces the product measure on $[0,1]^n$. Let $\Delta_{i_1,\cdots,i_k}$ be the set of points in $[0,1]^n$ whose co-ordinates of $i_1,i_2,\cdots$ and $i_k^{th}$ ...
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1answer
35 views

Pass to the limit under the sign of the integral of $f_{n}(x)$ [closed]

I'm trying to investigate following limit: $$ \lim_{n \to \infty}\int_{0}^{\frac{\pi}{2}}\frac{cos^{n}x}{1 + x^{3}}dx $$ I have a couple of questions 1. Is it possible to find the limit by the ...
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2answers
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The set of real numbers whose product is rational is Borel in $\mathbb{R}^2$

Let $A=\{(x,y)|xy\in \mathbb{Q}\}$ be a set in $\mathbb{R}^2$. Show that A is a Borel set, and find its Lebesgue measure $m_2(A)$. This is an exercise of the chapter of Fubini and Tonelli's theorem,...
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Why isn't it obvious, from the definition of the essential supremum, that $\lvert f \rvert \leq \|f\|_{\infty}, \ \mu-$a.e?

Let $(X,\mathcal{A},\mu)$ be a measure space, and $f \colon X \to \mathbb{C}$ an $\mathcal{A}$-measurable function. My texbook defines, for essentially bounded functions with respect to $\mu$, i.e., $...
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Density and Outer Measure

If $A \subset \mathbb{R}^{n}$ is Lebesgue measurable, then we see that $x \in A$ is a point of density if $$ \lim_{r \to 0} \frac{\mu(A \cap B(x,r))}{\mu(B(x,r))} =1 $$ The Lebesgue Density Theorem ...
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Convergence in measure and related inequality

Let $f_n$ be a sequence of bounded measurable functions on some probability space. Let $f_n$ does not converge to $0$ a.e. Does it mean that there exists a set of positive measure $E$ such that $\lim\...
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Smallest ring, algebra, $\sigma-$ring and $\sigma-$algebra in P(N) that contains Z:{{n}:n is even}

I need to find the smallest ring, algebra, $\sigma-$ring, and $\sigma-$algebra in P(N) that contain Z:{{n}:n is even}, where P(N) is the power set. I think that the smallest ring is {$\phi$,{0},{2},{...
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Proving implications between statements relating to Lebesgue-Stieltjes measure

$\mathcal{M}_\mu$ is the domain of the Lebesgue-Stieltjes measure $\mu = \mu_F$, where $F$ is an increasing, right continuous function on the Reals. I'm trying to prove the following: If $E \subset \...
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$(\int f_1d\mu)^2+\cdots+(\int f_nd\mu)^2\leq(\int \sqrt{f_1^2+\cdots+f_n^2}d\mu)^2$

Let $(X, \mathfrak{B}, \mu)$ be a measurable space, possibly not $\sigma$-finite, and $f_1, \cdots, f_n \colon X\to (-\infty, +\infty)$ be integrable functions on $X$. Does $$(\int f_1d\mu)^2+\cdots+(\...
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Equivalent notions of uniform integrability

I have seen many questions related to uniform integrability in this forum I though that it might be of interest to some to have yet another set of equivalent notions of uniform integrability. Here is ...
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1answer
40 views

PDF with a finite second moment converges to zero?

Suppose for a probability density function $p(x)$, $x\in \mathbb{R}$, we have $\int_{-\infty}^{+\infty} x^2p(x)dx < \infty$. Can we claim $\lim_{x\rightarrow\infty} p(x) = 0$? I think it boils ...
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Are jointly measurable adapted processes relative to natural filtration from right-continous processes, progressively measurable?

Recently i'm studying the book of Stroock and Varadhan - "Multidimensional Diffusion Processes" and i'm trying to solve this exercise : For the first part i argue in the following way : "Because ...
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Comparing the sizes of null sets

This question is about comparing the relative sizes of null sets by switching from open covers to open covering sequences (a la strong measure zero sets or microscopic sets). The main question is ...
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Show that pushforward measure is inner regular

Let $X, Y$ be compact Hausdorff spaces, $\tau:Y \to X$ continuous and $\mu^\tau:=\mu \circ \tau^{-1}$ the pushforward measure of the measure $\mu$ which is a inner regular measure. Show that $\mu^\tau$...
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If $Q$ is a rectangle and if $D\subseteq Q$ is the set of points where $f:Q\rightarrow\Bbb{R}$ is discontinuous then $\int_Q f$ exist if $m(D)=0$

What shown below is a reference from "Analysis on manifolds" by James R. Munkres So I want discuss why if $f$ is continuous at $a$ then there exist a rectangle $Q_a$ such that $a\in\overset{°}Q_a$ ...
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Expected value of random variable over a shrinking set. (Left-derivative of superexpectation)

I am working on a proof related to the left-derivative of the superexpectation operator $E_X(x) = E[\max\{X, x\}]$. Let $X \in L^2(\Omega, \mathcal{F}, P)$ be a random variable. Let $x', x \in \...
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Is $(X, Y)$ always absolutely continuous with respect to $P_X \otimes P_Y$?

Definitions: Let $X: (\Omega, \mathcal A) \to (\mathbb R, \mathcal B)$ be a random variable on the probability space $(\Omega, \mathcal A, P)$ and define its distribution as the probability measure $...
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If $f \in L^1[0,1]$ satisfies $\int_{0}^1f(x)e^{-2\pi i n x}dx = 0$ ; is $f$ zero a.e?

If $f \in L^1[0,1]$ satisfies $\int_{0}^1f(x)e^{-2\pi i n x}dx = 0$ for $n \in \mathbb{N}$ ; is $f$ zero a.e? Here $L^1[0,1]$ means the set of measurable functions $g : \mathbb{R} \rightarrow \mathbb{...
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Can we get simultaneous convergence of the integral using simple functions?

Let $Y$ be a nonnegative random random variable on some probability space $X$, and Let $F:[0,\infty) \to \mathbb R$ be a continuous function. Do there always exist simple functions $Y_n \ge 0$ on $...
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15 views

$ \int_{T}^{*}{\psi (t) d\mu(t)}=\int_{T}{\phi (t) d\mu(t)} $

Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space. The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by: $$ \...
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1answer
30 views

Sigma algebra generated by product of random variable

This question relates to this. Let $X,Y$ be two (real) random variables on $(\Omega,F)$. It was stated that $$\sigma(XY)\subseteq \sigma(X,Y).$$ The argument used is as follows: let $f(x,y)=xy$ be a ...
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$ \int_{E}^{*}{h(t,f_n(t)d\mu(t)}=\int_{E}{a_0\big(t,f_n(t)\big)d\mu(t)} $

Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space, and $E$ a convex subset of a Hausdorff topological vector space. Any function from $T\times E$ into $( - \infty, + \infty]$ is be called ...
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1answer
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How to prove this function is L-continuous almost-everywhere?

Definition: Let $(X, \mathcal{M}, \mu)$ a measure space. Some property P (in this case, the continuity) is said to be satisfied almost everywhere in X if there ...
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1answer
21 views

Inclusion of sigma-algebra generated by a sequence of random variables

Let $x_n$ be a sequence of random variables on $(\Omega,F,P)$ and $a_n$ be a sequence of real numbers. For any $1\leq i<k<\infty$, I want to evaluate if $$\sigma(a_mx_m: i\leq m\leq k)\subseteq \...
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29 views

The boundary of a disc has zero measure

Consider that a subset $A$ of $\mathbb{R}^{n}$ has measure $0$ if for every $\epsilon > 0$, there is a cover $\{U_{1}, U_{2}, \cdots \}$ of $A$ by closed (or open) rectangles such that $$\sum_{i = ...
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1answer
32 views

there exists a sequence of simple functions $\{g_n\}$ bounded in $L^1$, such that $f_n-g_n\to 0 \text{ a.e and in } L^1.$

Let $(E,\mathcal{A},\mu)$ be a finite measure space, and $\{f_n\}$ be a sequence bounded in $L^1$. Why does there exist a sequence of simple functions $\{g_n\}$ bounded in $L^1$, such that: $$ f_n-...
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1answer
44 views

Can we say that: $\|g\|_\infty<M $?

Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}\subset \mathcal{L}^2$, such that: $$ \forall n\geq 1~:~|f_n|\leq M~~a.e\qquad (1) $$ and $$ \exists g\in\mathcal{L}^1, \text{such that ...
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I can say that : according to the previous lemma we have: $ \| u_k \|_2\leq 2 \| F_k (f_n ^ k) \|_2, \qquad \forall n \geq 1.$

Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $ X $ be a Banach space and $ H $ a Hilbert space. For $ t \in E $, we set $ F_a (f)(t) = f (t) 1_{\| f \| \leq a} (t)$ Lemma: Let $ \{...
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All bounded sequence of $\mathcal{L}^2$ is weakly relatively compact of?

Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}\subset \mathcal{L}^2$, such that: $$ \sup_n \int_{E}{|f_n(t)|^2d\mu(t)}<+\infty $$ Why $\{f_n\}$ is weakly relatively compact of $\...
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29 views

$\sigma$-algebra generated by a finite family of sets

The question is the exercise 1.7 of "Measure theory and probability theory" by Krishna and Soumendra. Let $\mathcal{B}=\{B_1, ..., B_k\}$ with $B_i \subseteq \Omega$ and $\bigcup_{i=1}^kB_i=\Omega$. ...
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25 views

Reference to Lebesgue space on manifolds [closed]

I am looking for a reference to the Lebesgue space $L^p(M)$, where $M$ is a manifold.
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Finding adjoint operator

Let $X, Y$ be compact Hausdorff spaces. Let $\tau:Y\to X$ be continuous and $A \in \mathcal B(C(X),C(Y))$ the operator given by $Af:=f\circ \tau$. Show that the adjoint operator $A' \in \mathcal B(...
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Criterion for function to be measurable on product space

I was wondering if there is some sort of go to criterion to check if a function is measurable on a product space. Consider for example the following situation. Let $f: \Omega_1 \times \Omega_2 \to \...
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31 views

Asymptotically vanishing probabilities

Let $X_1,\ldots,X_n$ be a sequence of independent and identically distributed random variables, with joint probability measure $\mu^{(n)}$. Let $f_n:\mathbb{R}^n \to \mathbb{R}$ and $g_n:\mathbb{R}^n \...
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1answer
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sigma algebra created by partition

So there is a countable (disjoint) partition of $\Omega = \bigcup_{i \in \mathbb{N}}B_i$ and now I'm interested in the $\sigma$-algebra created by this partition $\sigma(\{B_i:i\in\mathbb{N}\})$. I've ...
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37 views

How to prove this equation with complex number?

I am stuck to proof this ineuqality. $|\frac{z}{z^2-1}-\frac{z}{z^2-4}|\leq\frac{\pi}{2\eta}$ with $\eta=Im z$ (Imaginary part of complex number). $Im z>0$ Here z is complex and $\epsilon\in(0,1)...
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1answer
24 views

Why does linear combination of dirichlet function is not a step function?

I know that Dirichlet function is not Riemann integrable but only Lebesgue integrable which has measure zero. I am having issue to understand the example that a simple function which is not step ...
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1answer
34 views

Can we say that $ \frac{1}{n}\sum_{i=1}^{n}{f_i(t)\to 0 }~~\text{a.e}$?

Let $(E,\mathcal {A},\mu)$ be a finite measure space and $\{f_n\}$ be a sequence bounded in $L^1$, such that: $$ f_n(t)\to 0 ~~\text{a.e and in } L^1 $$ Can we say that $$ \frac{1}{n}\sum_{i=1}^{n}{...
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19 views

Sigma-algebra generated by a sequence with zeros

Let $\{x_i\}_{i\in\mathbb{N}}$ be a sequence of zero mean random variables on a probability space $(\Omega, F, P)$. Suppose that there is some $i_0>0$ such that $x_i=0$ for all $i\geq i_0$. Does ...
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27 views

Doubt about the computation of the volume of a sphere

Consider the measure $d\Omega_{n+1}$ as a invariant measure induced by the Lebesgue measure on the unit sphere $\mathbb{S}\subset \mathbb{R}^{n+1}$. I want to calculate $d\Omega_{n+1}(\mathbb{S})$. I ...
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2answers
73 views

Is Banach–Tarski paradox false without axiom of choice? [duplicate]

I know that you need axiom of choice to prove Banach–Tarski paradox. But what happens with paradox when we remove axiom of choice? Does theorem become false? Or is there just no proof of it without ...
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24 views

Reference for the Dunford-Pettis Theorem.

Does anyone have a book reference for the Dunford-Pettis theorem as stated in the Wikipedia article https://en.wikipedia.org/wiki/Uniform_integrability.
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53 views

Hilbert space version of the notion of conditionally weakly-mixing functions.

$\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ $\newcommand{\mr}{\mathscr}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\E}{\mathbb E}$ $\newcommand{\C}{\mathbf C}$ ...
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49 views

Lp Space: an intuitive understanding.

Would someone be willing to give me an intuitive understanding of the Lp space? I have several analysis books which (seemingly) approach the topic differently, which confuses me more, and a simple ...
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14 views

Question regarding definition of Lebesgue integral

I am reading Measure Theory by Halmos, and I was wondering if someone could help me on the following definition: If there exists a mean fundamental sequence $\left\{f_{n}\right\}$ of integrable ...
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1answer
35 views

Is there a missing condition in this statement regarding Borel measurability

I'm self-studying measure theory. I stumbled upon this statement: If $X \subset \mathbb{R}$ and $f: X \to \mathbb{R}$ is a function, then $f$ is a Borel measurable function if and only if $f^{-1}((...

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