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

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Is $\sigma(\sigma(E)) = \sigma(E)$, where $\sigma(E)$ is the $\sigma$-algebra generated by $E$? [closed]

Given a space (set) $X$ and some subset $E \subset {\rm P}(X)$ of the power set of $X$. Is the sigma algebra generated by the sigma algebra generated by $E$ equal to the sigma algebra generated by ...
2
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0answers
56 views

Notation $x^n=(x_1,\dotsc,x_n)$

In a book on statistics I saw the notation $x^n=(x_1,\dotsc,x_n)$ and wondering how common this is in measure theory/statistics. More precisely it is about a probability space ...
3
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1answer
52 views

Approximating $L^p$ functions using Schwartz functions with compact support on the Fourier side

For $1\leq p<\infty$, how would you show for any $f\in L^p(\mathbb{R})$ and given $\epsilon>0$, there exists $L<\infty$ and $g\in \mathcal{S}(\mathbb{R})$ such that $\|f-g\|_p<\epsilon$ ...
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0answers
22 views

Prove that G is absolutely continuous and not monotone on any subinterval of $[0,1]$

Let $A\subset [0,1]$ be a Borel set such that $0<m(A\cap I)<m(I)$ for any subinterval $I$ of $[0,1]$. Let $G(x)=m([0,x]\cap A)-m([0,x]\setminus A)$. Prove that G is absolutely continuous and not ...
5
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0answers
40 views

A very quick way to prove a set is measurable. [duplicate]

All examples of non-measurable subset of $\mathbb{R}$ (in the Lebesgue sense) seem to need the axiom of choice in some way or the other. Hence, can we say: The set $A\subseteq \mathbb{R}$ is ...
2
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1answer
35 views

Breaking a Function in $L^{\infty}[0,1]$

Let $f\in L^{\infty}[0,1]$ s.t. $\|f\|_{\infty}=1$ $E:=\{x\in[0,1]:|f(x)|<1\}$ If $m(E)>0$, then is it possible to find $g,h\in L^{\infty}[0,1]$ such that ...
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0answers
16 views

Borel sets and absolutely continuous functions - second part

Borel sets and absolutely continuous functions This question is a part of the question of this link. So, in order to show that $F'=0$ on set of positive measure, what I did was mentioned here: Let ...
3
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1answer
43 views

Borel sets and absolutely continuous functions

Let $A\subset [0,1]$ be a Borel set such that $0<m(A\cap I)<m(I)$ for any subinterval $I$ of $[0,1]$. Let $F(x)=m([0,x]\cap A)$. Show that $F$ is absolutely continuous and strictly increasing on ...
1
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0answers
32 views

Amalgamated product of two measures.

Please do not get annoyed by the symbols below. The problem has a really simple statement. $2^X$ denotes the set of functions from $X$ to $\{0, 1\}$ equipped with usual product topology. By $Fn(X, ...
4
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0answers
49 views

There exists a Borel set such that $0<m(A\cap I)<m(I)$ for every interval of $[0,1]$. [closed]

Is this seems so odd to me, can someone please help? I think it should be some Cantor type construction but can't figure out.
2
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1answer
12 views

If $\mathcal A$ generates $\mathcal S$ then $\sigma (X )=\sigma (X ^{-1 } ( \mathcal A ))$

Show that if $\mathcal A$ generates $\mathcal S$ then $X ^{-1 } ( \mathcal A )= \{\{X \in A \} : A \in \mathcal A \} $ generates $\sigma (X) =\{\{X \in B \} : B \in \mathcal S \}$ First I ...
1
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1answer
21 views

Lebesgue integrable discontinuity points

If a function is Lebesgue integrable, is it possible that it has as set of discontinuity points measure bigger than zero?
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1answer
19 views

How do I finish the proof or make this more rigorous? Outer measure

These are supposed to be a one-liner proof, but I can't make it rigorous for some reason. Find $m^*(rE)$ in terms of $m^*(E)$ where $rE = \{rx: x \in E \}$. Prove $m^*(E\cup F) \leq m^*(E) + ...
2
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2answers
46 views

How to use dominated convergence theorem?

How to use dominated convergence theorem to compute $$lim_{n\rightarrow \infty}\int_0^1\frac{1+nx^2}{(1+x^2)^n}$$ So far I have only done $\frac{1+nx^2}{(1+x^2)^n}\le\frac{1+nx^2}{(1+x^2)}$, I don't ...
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0answers
15 views

Lebesgue Outer Measure of Unbounded Interval

I am reading from Carothers' Real Analysis. He defines the outer (Lebesgue) measure of a subset $E$ of $\mathbb{R}$ as: $$m^*(E) = \mathrm{inf}\left\{ \sum_{n=1}^{\infty} l(I_n) L : E \subset ...
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2answers
29 views

Show that it stands for each measurable $E$

Let $f$ be integrable in a space of finite measure. Show that $\forall \varepsilon >0$ $\exists \delta>0$ such that for each measurable $E$ with $\mu(E)<\delta$ we have that $$\int_E|f|d\mu ...
0
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2answers
30 views

integration with respect to signed measure

Let $\mu$ be a signed measure. For positive $\mu$, we know that $$\left|\int f \, d\mu \right|\leq \int| f| \, d\mu$$ Is this still true if we drop the positivity assumption?
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1answer
39 views

Precise meaning/implications of “a random variable belongs to a space” almost surely.

As far as I understood, by saying a random variable/vector $X$ belongs to a space $S$ (or takes values in $S$), one means that the measurable function $X$ is $S$-valued: \begin{equation} ...
3
votes
2answers
60 views

Showing that $f = 0$ a.e.

Let $f$ be a real-valued, Borel measurable and integrable function on $[0,1]$. Suppose that $\int_A f = 0$ for all Borel subset $A$ of $[0,1]$ with measure $\frac12$. Prove that $f = 0$ a.e. This is ...
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2answers
40 views

Finer partitioning of the $[0,1]$ interval

Let $S\subset [0,1]$ . Lebesgue Density Theorem tells us that the density of $S$ is either $0$ or $1$ almost everywhere on the interval $[0,1]$. Define a $01$-transition to be a point $x\in [0,1]$ ...
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0answers
40 views

What is a convolution kernel?

What is a convolution kernel? (in measure theory, probability theory) In which book can I read about kernels on measurable spaces and convolution kernels? Thank you!
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1answer
47 views

Series representation of measurable functions

I need to show that if a positive function $f$ on $E$, where $(E,\mathcal{E})$ is a measurable space, is $\mathcal{E}$-measurable then it has the form $$f=\sum_{n=1}^{\infty}a_n1_{A_n},$$ for some ...
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2answers
32 views

Integrability: Cauchy Sequence

This thread is related to: Spectral Measure: Dominated Convergence Given a measure space $\Omega$. Consider a sequence of square integrables: $\int|f_n|^2\mathrm{d}\mu<\infty$ Suppose pointwise ...
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2answers
37 views

Show that $(E\cup Z_1)\setminus Z_2$ has the form $E\cup Z$

I'm trying to do an exercise as follows: Let $(X, {\mathbf X}, \mu)$ be a measure space and let ${\mathbf Z}=\{E\in {\mathbf X}:\mu(E)=0\}$. Let $\mathbf X'$ be the family of all subsets of $X$ of ...
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1answer
23 views

Cofinite Topology: Borel Algebra vs. Power Set

Being curious... Are there uncountable spaces such that any uncountable subset has countable complement: ...
3
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1answer
57 views

A function constant almost everywhere

Q: Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is measurable with respect to Lebesgue measure and $f(x)=f(x+1)=f(x+\pi)$ for almost every $x$. Prove that $f$ is constant almost everywhere. Proof ...
3
votes
1answer
36 views

Unique measure on positive Borel sets with following properties

This is actually part (a) question asked previously. However, that person had solved part (a) and so no solution is available. I am trying to prove it myself but am running in to difficulties. ...
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0answers
21 views

Continuity of set function on field and relation with continuity in topological space

I am trying to understand how continuity of measures relates to the definition of continuity in topological sets : Every open set in range corresponds to an open set in domain. A real valued set ...
3
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1answer
49 views

Increasing functions on $\mathbb{R}$

If $F$ is increasing on $\mathbb{R}$ then show that $F(b)-F(a)\geq \int_a^b F'(t)dt$. My work: Since $F$ is increasing on $\mathbb{R}$, $F'$ exists a.e. on $\mathbb{R}$. So $F'$ is integrable on ...
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1answer
25 views

Integral of the indicator function of a measurable set with respect to the product measure.

Let us say that $E$ is a measurable set with respect to the product measure $\mu \times \nu$ on $X \times Y$. My professor uses the following, $$\int_{X \times Y} \chi_{E}(x,y) d( \mu \times \nu) = ...
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0answers
77 views

A problem with kernels on measurable spaces

Let $(E, \mathcal{B}(E)), (F, \mathcal{B}(F))$ be two measurable spaces. A $kernel$ from $(E, \mathcal{B}(E))$ to $(F, \mathcal{B}(F))$ is an application $N : p \mathcal{B} (E) \rightarrow p ...
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1answer
32 views

Show that: $ X \ {\text{is}}\ \mu{\text{-integrable}} \implies \sum_{k=1}^\infty\mu(\{\mid X\mid ≥ k\}) < \infty$

Assignment: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $X: \Omega \rightarrow \bar{\mathbb{R}}$ a $\mathfrak{A}$-$\bar{\mathfrak{B}}$-measurable function. Show that: $$ X \ ...
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2answers
32 views

Show that: $\int_{\Omega} X d\mu = 0 \iff \mu(\{\omega\in\Omega\mid X(w)>0\})=0$

Assignment: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $X: \Omega \rightarrow \bar{\mathbb{R}}$ a non-negative $\mathfrak{A}$-$\bar{\mathfrak{B}}$-measurable function. Show that: ...
0
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1answer
41 views

Integral equality over $\mathbb{R}$ for measurable functions in $\mathbb{R}^{n}$.

I originally asked a question about notation, but in view of PhoemueX's thoughtful answer, I have edited the title/question statement so that this question can more searchable in the future. Given a ...
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1answer
26 views

Prove $f(x) = \frac 1 {\sqrt{2\pi}} \int_{\mathbb R} \hat f(t) e^{itx} \ \lambda(dt)$ for every $x \in \mathbb R$.

Let $\mathcal L_{\mathbb C}^1(\lambda)$ such that $\hat f \in L_{\mathbb C}^1(\lambda)$ (Fourier transformation). I've proven that $f(x) = \frac 1 {\sqrt{2\pi}} \int_{\mathbb R} \hat f(t) e^{itx} \ ...
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0answers
16 views

If $\mu'$ denotes the pushforward measure, then $\int f\circ X\;d\mu=\int f\;d\mu'$

Let $(\Omega,\mathcal{A})$ and $(\Omega',\mathcal{A}')$ be measure spaces $\mu$ be a measure on $(\Omega,\mathcal{A})$ and $\mu':=\mu\circ X^{-1}$ be the pushforward measure of $\mu$ under $X$ ...
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0answers
37 views

Show $\int_{\mathbb R} f * g \ d \lambda = \int_{\mathbb R} f \ d \lambda \cdot \int_{\mathbb R} g \ d \lambda$.

Suppose $f,g \in \mathcal L_{\mathbb C}^1(\lambda)$. Show $\int_{\mathbb R} f * g \ d \lambda = \int_{\mathbb R} f \ d \lambda \cdot \int_{\mathbb R} g \ d \lambda$. I see that $\int_{\mathbb R} ...
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0answers
64 views

A special property for Lebesgue measurable sets.

A defined measure $\mu$ on the Lebesgue measurable sets in $\mathbb{R}^N$ holds the property $\star$ if $\mu(K)<\infty$ for each $K$ compact in $\mathbb{R}^N$ and if exists constant $C>1$ such ...
2
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1answer
31 views

A basic question on $L^p$ norm

Consider a probability space and $f_m$ be sequence of measurable functions a.s. converging to $f$. What can be said about the limit $$ \lim_{m\to \infty} \|f_m\|_m$$ where $\|.\|_p$ stands for the ...
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1answer
35 views

How to give a good upper bound on tail probability for $P\{|\frac{R_n}{\sqrt{n}}-1| \ge \varepsilon\}$?

Suppose $X_1,X_2,\ldots$ is a sequence of i.i.d. standard normal random variables. $R_n=\sqrt{X_1^2+\ldots+X_n^2)}$. How could I prove $P\{|\frac{R_n}{\sqrt{n}}-1| \ge ...
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1answer
56 views

Unifying the treatment of discrete and continuous random variable

I have been working on the reconciliation of the treatment of discrete and continuous random variable in a measure theoretic sense. But I found myself blocked on fundamental results. We know that If ...
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0answers
35 views

Completeness of the space of all convergent sequences in $\mathbb{R}$. [duplicate]

Let $X$ the set of all real-valued convergent sequences. Consider the space $(X, \Vert \cdot \Vert_{\infty})$. Is it complete?
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12 views

Let $B$ be the Banach space of complex measures, with norm $|\mu|(X)$. What is the Banach space interpretation of notions such as mutual singularity?

Let $B$ be the Banach space of complex measures, with norm $|\mu|(X)$. What is the Banach space interpretation of measure theoretic notions such as mutual singularity? Is there a useful dictionary ...
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2answers
43 views

Why does it imply from that, that $f=0$ almost everywhere?

When $\int_G f=0, \forall$ open $ G$, we have that if $E\subset G$ then $\exists E \subset G$ such that $m(G \setminus E)< \epsilon$. So, $$\int_G f=\int_E f+\int_{G \setminus E } f$$ $\int_G ...
1
vote
1answer
89 views

Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
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0answers
11 views

Coarea formula for fractional dimension

The coarea formula states that any locally Lipschitz function (e.g. a $C^1$-function) $F:\mathbb{R}^N\to\mathbb{R}^n$ with $N\geq n$ satisfies $$\int_A JF(x) \mathrm{d}\mathcal{H}^N = ...
2
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1answer
40 views

What does it mean u(dx) in the Fourier transform of a probability measure u?

Let $\mu$ be a probability on $\mathbb{R}^n$ and consider its Fourier transform $\overset{\wedge}{\mu} (u) = \int e^{i (u ,x)} \mu( dx)$, where $(u, x)$ is the scalar product of $u$ and $x$. What ...
2
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0answers
30 views

Showing inequalities for $l^p$ sequences

If I show that an inequality (e.g. Holder or Minkowski) holds for the $L^p$ space, then can I automatically conclude that the inequality also holds for $\ell^p$ sequences, just by integrating wrt. the ...
1
vote
1answer
55 views

Relying two points by an almost-geodesic omitting a singular set a.e.

I failed to give an appropriate title to the question, so any suggestion for a better title is welcome: Here's the question: I would like to prove the following result: Given $\varepsilon>0$, a ...
0
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2answers
60 views

Show that this is the limit

In a space with measure $1$, $||f||_p$ is a oncreasding function with respect of $p$. To show that $\lim_{p \rightarrow \infty} ||f||_p=||f||_{\infty}$ we have to show that $||f||_{\infty}$ is the ...