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

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Prove that every set that is content zero is also measure zero.

Prove that every set that is content zero is also measure zero. I understand that this is true, but am not sure how exactly to prove it.
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1answer
51 views

Why does $\sigma (X_t) \subset \sigma (X)$ hold?

We have a random process $X=\{X_t\;,t\in T\}$, where $X_t:(\Omega,\mathcal{A})\to(S_t,\mathcal{S}_t)$ are random variables. I am confused as to why does $$\sigma (X_t) \subset \sigma (X)$$ hold ...
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Lebesgue Integral: Convexity

Given a probability measure $\rho(\Omega)=1$. Consider a complex function $f\in\mathcal{L}(\rho)$. From the Riemann integral it is evident that: $$\int_\Omega f\mathrm{d}\rho\in\overline{\langle ...
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1answer
25 views

Complex Measures: Jacobian

Disclaimer This thread is meant to record results and written as jeopardy. ;) (For more details see: Answer own Question) Problem This is a follow-up to: Borel Measures: Jacobian Given a complex ...
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1answer
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Borel Measures: Jacobian

Disclaimer This thread is meant to record results and written as jeopardy. :) (For more details see: Answer own Question) Problem Given a sigma-finite measure $\lambda$. Consider a finite measure ...
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18 views

Borel Algebra: Nonexample?

Are there any sigma-algebras not induced by some topology? (I'm thinking of something over finite space but not sure wether it must be infinite.)
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Definitions of Lebesgue integral

I know the definition, from Kolmogorov and Fomin's Элементы теории функций и функционального анализа, of Lebesgue integral of measurable function $f:X\to \mathbb{C}$ on $X,\mu(X)<\infty$ as the ...
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Show that $\lim _n (X \setminus E _n ) =(X \setminus \lim _n E _n )$, for $E_n $ an increasing sequence of sets

Let $\{E _n \} $ be a sequence of subsets of $X $ defnined as $E _n=\cap _{m=n } ^{\infty } A _m $, so that $E _n \subset E _{n+1 }\subset ...$ . Let $E = \lim E _n = \cup _{n=1 } ^{\infty } E _n$ I ...
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1answer
21 views

An inequality for functions in $L^1$ and $L^2$.

Given $f\in L^1(X)$, show that for every $\rho>0$ one has: $\mu({|f|>\rho})\leq \rho^{-1}\int|f|d\mu$. I think in order to prove this, we use the fact that $\int|f|d\mu= \lim_{\rho\to ...
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1answer
39 views

Continuity from property of constriction images of spheres

Let $D\subset\mathbb R^n$ --- domain and mapping $\varphi:D\to \mathbb R^n$. The following property holds There is a set $T\subset D$ s. t. measure $|D\setminus T|=0$ and for every point ...
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1answer
29 views

Product measures and $\sigma-$ finite measures

Problem similar to folland chapter 2 problem 51. The actual problem in Folland mentions that $X,Y$ are not necessarily $\sigma-$finite. Then how can I use Fubini-Tonelli theorem?
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Show that $f=0$ almost everywhere.

If $f$ is integrable in $\mathbb{R}^d$ as for the Lebesgue measure and $\int_{R}f=0$ for each rectangle $R$, then $f=0$ almost everywhere. Could you give me some hints how to show it??
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0answers
26 views

Product measures in measure theory

Suppose $(X,M,\mu)$ and $(Y,N,v)$ are $\sigma-$finite measure spaces. Is the following calculation correct? $\int \chi_E(x,y) d\mu=\int (\chi_E)^y(x) d\mu=\int \chi_{E^y}(x) d\mu=\mu(E^y)$ for a ...
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Hausdorff measures and densities

I've been stuck on this one for a while now. It's problem 2.4 from Falconer's "The geometry of fractals" Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with ...
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1answer
42 views

What is this theorem about measurable functions saying?

Theorem: Let $(\Omega,\mathcal{F})$ be a measurable space and let $f:\Omega \rightarrow Y$ be a given function. Let $\mathcal{A}$ be a collection of subsets of $Y$. If $f^{-1}(A) \in \mathcal{F}$ ...
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1answer
27 views

Convergence in $L_p$ and elsewhere

Let $\|f\|_p:=(\int_X|f|^pd\mu)^{1/p}$ and let $L_p$ be the space of (the classes of equivalence of) complex or real measurable functions such that $\int_X|f|^p d\mu<\infty$ exists. In ...
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1answer
29 views

Sequence that converges as for the norm but not almost everywhere

How can I find a sequence that converges as for the norm but doesn't converge almost everywhere, in some space $L^p$ ?? Could you give me some hints ??
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58 views

Counterexample to see consequence of monotone convergence theorem

Does there exist an increasing sequence of functions (fn) converging to a function f and a sequence of integrable functions (gn) such that fn>= gn for all n. And (gn) also converges to a function g ...
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2answers
25 views

Does L1 and nonnegative imply bounded almost everywhere?

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a nonnegative function, such that $f\in L^1(\mathbb{R})$. Does this imply that $f$ is bounded almost everywhere?
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Convergence in a Measurable set

Let $E$ the set of all $x\in[0,2\pi]$ at which $\{\sin (nx)\}$ converges. This implies that $E$ is measurable? Thanks you all.
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Finding countable compact set s.t $\underline{\dim}_M(K)\lneq\overline{\dim}_M(K)$

Im trying to find a countable compact set such that $$\underline{\dim}_M(K)\lneq\overline{\dim}_M(K)$$ I tried thinking about Koch curve, sierpinskii gasket and carpet, Bedford-McMullen carpet and ...
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1answer
19 views

Smith-Volterra-Cantor set remove “m”th interval

Is it possible to determine the measure of the Cantor set by removing the middle "m"th interval (m=1,2,3,4,...) from [0,1]? For example, removing middle 3rd from [0,1] gives measure 0; removing ...
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2answers
34 views

Union of sets as the union of disjoint sets - Does the proof $\forall n\in \mathbb{N}$ implies the proof for infinity?

I managed to prove that: $$\displaystyle\bigcup_{i=1}^n A_i=A_1\cup(A_1^c\cap A_2)\cup(A_1^c\cap A_2^c\cap A_3)\cup\dots\cup(A_1^c\cap\dots\cap A_{n-1}^c\cap A_n)$$ for $\forall n \in\mathbb{N}$. ...
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41 views

Relationship between measure theory and real analysis

Does measure theory generalize real analysis to abstract spaces? For example, you can now talk about convergence even on unordered fields.
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46 views

Assumptions involving product spaces

Suppose a random variable $X$ is distributed in $\mathbb{R}^{n}$ and we are given that $X' = (X_{1}', X_{2}')$ for $X_{i}$ distributed on $\mathbb{R}^{n_{i}}$. In general, what assumptions can I make ...
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1answer
38 views

$M\subset\mathbb{R}^n$ measurable. Show: There is a null set $N \subset \mathbb{R}^n$ and compact seq $K_m$ with $M=N\cup\bigcup_{m\in\mathbb{N}}K_m$.

Assignment: Let $M\subset\mathbb{R}^n$ be lebesgue-measurable. Show that, there is a null set $N \subset \mathbb{R}^n$ and a sequence $(K_m)_{m\in\mathbb{N}}$ of compact subsets $K_m \subset ...
3
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1answer
41 views

If $B\subset\mathbb{R}^n$ is measurable and $(x_l)_{l\in\mathbb{N}}$ is a bounded family, so $(B + x_l)$ is pairwise disjoint, then $\mu(B)=0$.

Assignment: Show that: If $B\subset\mathbb{R}^n$ is Lebesgue-measurable and if there is a bounded family $(x_l)_{l\in\mathbb{N}} \subset \mathbb{R}^n$ so that the family $(B + ...
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1answer
43 views

Bounding $P(X \le \tau)$

I am trying to upper bounding $P(X \le \tau)$ where $X$ is non-negative r.v. and where $\tau \le 1$. I have become aware of the Reverse Markov inequality that says that, if $P(|X|\le a)=1$ then for ...
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1answer
34 views

$L_2$ as a Hilbert space and $\ell_2$

I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where ...
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1answer
52 views

Help in a problem about Lebesgue integration inequality

Let $ (X,\mathcal{S},\mu)$ be a finite measure space, let $f$ be $\mathcal{S}$-measurable and let $E_{n}:= \{x\in X :n-1\le |f(x)|<n\}$ for $n=1,2,\dots$ Show that: $$f \in ...
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71 views

Soft Question: Are sigma fields, fields?

I'm sorry if this is a foolish question but: Is a $\sigma$-field (of sets) a field (in the sense of algebra) if we only consider finite intersections and finite unions?
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1answer
27 views

every $\sigma$ algebra is a monotone class

I couldn't understand the monotone class theorem because of this lemma: "Every $\sigma$ algebra is a monotone class." How i can prove it?
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given $E_1, E_2, E_3, …$ prove that the measure of {$x \in X :$ x belongs to infinite number of sets $E_k$} is $0$

Say I have a $\sigma$-algebra $\mathcal{A}$ over a set $X$ and a measure $\mu$. Let $E_1, E_2, E_3, .... \in \mathcal{A}$ such that $\sum_{k=1}^\infty \mu(E_k)$ < $\infty$. let B = {$x \in X ...
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1answer
19 views

If $f(\cdot,y)$ is measurable and $f(x,\cdot)$ is continuous, $\{x:|f(x,y)-f(x,0)| \leq \epsilon, \; \;\forall y <\delta\}$ is measurable

Suppose $\mu(X) < \infty$ and $f : X \times [0,1] \rightarrow \mathbb{C} $ is a function such that $f(\cdot,y)$ is measurable for each $y \in [0,1]$ and $f(x,\cdot)$ is continuous for each $x ...
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1answer
26 views

Example of algebra that is not a $\sigma$-algebra

I understand that an algebra $F \subset 2^\Omega$ is called a $\sigma$-algebra if it additionaly satisfies: $(A_i)_{i \in \mathbb{N}}$ with $A_i \in F$ pairwise disjoint, then also $\cup_{i \in ...
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1answer
27 views

Use the Monotone Convergence Thm, to show $\displaystyle\int f \le \liminf \int f_n$

! (http://i.imgur.com/Zwt1m1n.png) I need to do the question at the top of this image. I figured out that $g_n$ is an increasing sequence that is pointwise convergent to $f$. i.e. I know $\lim ...
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1answer
27 views

Where does the following intuition about $G_\delta$ sets fail?

Where does the following reasoning that $\mathbb{Q}$ is supposedly a $G_\delta$ set fail? "Proof": $\mathbb Q$ may be covered by selecting open sets $O_n$ such that $m(O_n)<\frac{1}{n}$ for ...
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Basic measure theory: Composite functions and points of nondifferentiability

I have a function $V(x(t))$. $x(t)$ is continuous, but not everywhere differentiable w.r.t. $t$. What can we say about $V$ at these points of non-differentiability? To explain I have included some ...
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A question about inclusion of $L^r(\mu)$ spaces for different $r$ and different measures $\mu$

For some measures, the relation $r<s$ implies $L^r(\mu)\subseteq L^s(\mu)$ ; for others, the inclusion is reversed; and there are some for which $L^r(\mu)$ does not contain $L^s(\mu)$ if $r\ne ...
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3answers
96 views

convolution of characteristic functions

Suppose $A$ and $B$ are measurable subsets of $\mathbb{R}$ of finite positive measure. Show that the convolution $\chi_A*\chi_B$ is continuous and not identically $0$. Use this to prove that $A+B$ ...
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“Uniform” Convergence in Distribution (bounded Lipschitz metric)

I have been thinking about the following problem. Let me know if the notation below makes sense. Let $\mathcal{P}$ denote the set of Borel probability measures on a metric space $(\mathbb{R}^{k}, ...
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Billingsley Probability and Measure Solution Manual?

Does anyone know where I can get a solution manual for Billingsley Probability and Measure? I am doing a self study and need help understanding a couple problems. Thank you.
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1answer
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Exercise on measure theory, (verification and suggestion)

Hi everyone I'd like to know if the following is correct and also I'd appreciate any suggestion to improve the argument. Thanks in advance For every positive integer $n$, let $f_n:{\bf{R}}\to ...
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1answer
27 views

Limit of integral over set measurable

If $A\subset[0,2\pi]$ is measurable, prove that $$\lim_{n\to\infty}\int_A \cos (nx)\ dx=\lim_{n\to\infty}\int_A \sin(nx) \ dx=0$$ Please, any suggestions are welcome.
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45 views

Approximating simple summable function in measure space with countable base

Let $f:X\to \mathbb{Q}+i\mathbb{Q}\subset\mathbb{C}$, $f\in L_1(X,\mu)$ be a Lebesgue-summable function taking only finitely many values $y_1,\ldots,y_n\in \mathbb{Q}+i\mathbb{Q}$ on the sets ...
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Product of Absolutely Continuous Measures is Absolutely Continuous

I am stuck on this problem from Folland's Real Analysis, Second Edition: For $j = 1, 2$, let $\mu_j, \nu_j$ be $\sigma$-finite measures on $(X_j, \mathcal{M}_j)$ such that $\nu_j <\!\!< \mu_j$. ...
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1answer
46 views

Showing Convergence

Let $(X,M,\mu)$ be a measurable space and $f$ be a real valued integrable function on $X$. Let $E_n=\{x\in X: f(x)\geq nq\}$ for every $n\in \mathbb{N}$ and fixed $q>0$ . Show that ...
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1answer
19 views

(Hints please) Constructing a measurable set with the following property.

If $\delta >0$, $I_\delta=(-\delta,\delta)\in\mathbb{R}$, and $0\leq\alpha\leq\beta\leq1$, what hints do you have that would help me figure out how to construct a measurable set ...
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1answer
11 views

Showing Convergence in measure with some condition. [closed]

Let $(X,M,m)$ be a finite measurable space and $\{f_n\}$ be a sequence of real valued measurable functions on $X$ . Let $$E_n=\{x\in X : f_n(x)\ne 0\}$$ for every $n\in \mathbb{N}$ . Show that if ...
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2answers
26 views

Integral Measure: Total Variation Preservation

Given a measure $\lambda$. Consider a real measure $\mu(E):=\int_E h\mathrm{d}\lambda$. Then its total variation measure is given by: $$|\mu|(E)=\int_E|h|\mathrm{d}\lambda$$ How to prove this? (I'm ...