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

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1answer
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Continuity and convergence everywhere.

Suppose a sequence of continuous functions $(h_n)$ converges almost everywhere to another continuous function $h$ . Is it possible to infer that $h_n$ infact converges everywhere? If not, under what ...
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0answers
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Question with Lebesgue-Stieltges outer measure

Let $g$ and $h$ be two increasing functions and $\theta_g$, $\theta_h$ be the associated Lebesgue-Stieltges outer measures on $R$ (the set of real numbers). We can also associate to $g+h$ the L-S ...
3
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1answer
45 views

Real Analysis, Folland Proposition 3.11 The Radon Nikodym Theorem

Proposition 3.11: If $\mu_1,\ldots,\mu_n$ are measures on $(X,M)$, there is a measure $\mu$ such that $\mu_j\ll\mu$ for all $j$ -- namely $\mu = \sum_1^n\mu_j$. Attempted proof: Suppose we have a ...
3
votes
2answers
42 views

Real Analysis, Folland Corollary 3.10 The Lebesgue Radon Nikodym Theorem

Background Information: Proposition 3.9 - Suppose that $\nu$ is a $\sigma$-finite measure and $\lambda$ are $\sigma$-finite measures on $(X,M)$ such that $\nu\ll \mu$ and $\mu\ll \lambda$. a.)...
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2answers
14 views

Dealing with a Sequence of Sets with Two Indices and Simple Function based on that Sequence of Sets

Okay so I have a measurable function $f$ and a set $E_{n,i}$ $$E_{n,i}=\left\{ x:\frac{i-1}{2^{n}} \leq f(x)<\frac{i}{2^{n}}\right\}$$ where $i=1,...,n2^{n}$ and $n=1,2,...$ Then I have another ...
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0answers
33 views

Hölder inequality application to show that f=1

I want to proof that if $f \in L^{1}_{\mu}(\mathbb{R}), f > 0$ continuous, satisfies $(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
4
votes
1answer
104 views

Conclusion about measurable functions from knowledge about continuous functions

Let $\mu$ and $\nu$ be two finite Borel measures on $\mathbb{R}$. We know that if $$\int f d\mu = \int f d\nu $$ for all continuous functions $f$ then $\mu=\nu$ and so the equation above holds for ...
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0answers
14 views

Measure induced by subgradient of convex functional

I am trying to understand why the following defines a measure. Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function. Define a measure $\mu$ on $\mathcal{B}(\mathbb{R}^d)$ by $$\mu(E) = \...
4
votes
1answer
66 views

Lebesgue integration by substitution

I read that, if $f\in L^1[c,d]$ is a Lebesgue summable function on $[a,b]$ and $g:[a,b]\to[c,d]$ is invertible and such that $g\in C^1[a,b]$ and $g^{-1}\in C^1[a,b]$, then $$\int_\limits{g([a,b])}f(x)\...
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1answer
20 views

the relation between the sigma-algebras of two isomorphic spaces [closed]

It crosses my mind the following question : if X and Y are two isomorphic spaces what can we say about the Borel sigma-algebras associated to each of them, otherwise what is the relation between $\...
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0answers
16 views

$p$-power integral and $p$-series in higher dimensions

This seems like a basic question that should be addressed in a multivariable calculus course however I don't think I've ever confronted the issue until I became confused about a recent question here ...
6
votes
4answers
889 views

Is diameter of a set a measure?

Suppose the diameter of a nonempty set $A$ is defined as $$\sigma(A) := \sup_{x,y \in A} d(x,y)$$ where $d(x,y)$ is a metric. Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable ...
1
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1answer
29 views

Measure and probability : what would be $\mu(A\mid B)$?

I know that a probability space $(\Omega ,\mathcal F,\mathbb P)$ is in fact a measure space with a $\sigma -$algebra $\mathcal F$ and a measure $\mathbb P$. I know that if $A,B\in \mathcal F$ with $\...
0
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1answer
57 views

Real Analysis, Folland Corollary 3.6, The Lebesgue-Radon-Nikodym Theorem

Background Information: This is a Corollary to Theorem 3.5, found here. If $\mu$ is a measure and $f$ is an extended $\mu$-integrable function, the signed measure $\nu$ defined by $\nu(E) = \int_{E}...
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1answer
36 views

A function is Borel iff $f^{-1}(a,\infty) \in \mathcal{F}$

I'm reading Shao's Mathematical Statistics and part of a proposition is that, if $(\Omega, \mathcal{F})$ is a measurable space then: A function $f$ is Borel iff $f^{-1}(a,\infty)\in \mathcal{F}$ for ...
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2answers
30 views

How do you rigorously explain the fact that $u \in L^p$ can be non defined over sets of measure 0?

In all the definitions of $L^p(\Omega)$ spaces I have been given these are defined to be the set of functions $f: \Omega \to \mathbb{R}$ whose norm $||\cdot||_{L^p}$ is finite. We define is as the ...
2
votes
1answer
23 views

Cardinality of a borelian

My advisor told me that a Borel set can only be finite, countably infinite or having the cardinality of the continuum (obviously we are not assuming Continuum Hypothesis). I think he mentioned "...
1
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1answer
55 views

Showing region between $x$ axis and graph is in ${\cal{B}}(\mathbb{R^2})$

Let $(\mathbb{R^2},{\cal{B}}(\mathbb{R^2}),\lambda^2)$ be a measure space where $\lambda^2$ is the two dimensional Lebesgue measure. Let $u:\mathbb{R}\to [0,\infty]$ be a Borel measurable function. ...
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1answer
44 views

Real Analysis, Folland Theorem 3.5, absoulute continuity

Background Information: Suppose that $\nu$ is a signed measure and $\mu$ is a positive measure on $(X,M)$. We say that $\nu$ is absolutely continuous w.r.t. $\mu$ and write $$\nu \ll \mu$$ if $\nu(E) ...
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4answers
40 views

Can anyone explain the convolution of independent random variables?

Specifically, suppose that $X_1$ and $X_2$ are independent and $f$ is a Borel-measurable function s.t. $f(x_1, x_2)=\begin{cases} 1, & \text{if } x_1+x_2 \leq x\\ 0, & \text{otherwise} \end{...
0
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0answers
28 views

Inner lebesgue measure definitions

I´m currently studying lebesgue measure theory. I'm using Lebesgue integration on euclidean space by Frank Jones as one of my reference books. He defines the outer and inner lebesgue measures for an ...
6
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1answer
49 views

Example of set, finite outer measure, subsets, where outer measure does not converge

What is an example of a set $X$ and a finite outer measure $\mu^*$ on $X$, subsets $A_n \uparrow A$ of $X$, and subsets $B_n \downarrow B$ of $X$ such that $\mu^*(A_n)$ does not converge to $\mu^*(A)$ ...
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0answers
40 views

Does $\mu^*$ agree with $\mu$, measure space? [closed]

If $(X, \mathcal{A}, \mu)$ is a measure space, define$$\mu^*(A) = \inf\{\mu(B) : A \subset B,\,B \in \mathcal{A}\}$$for all subsets $A$ of $X$. I have a few questions? Is $\mu^*$ an outer measure? ...
0
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0answers
23 views

JL Doob / KL Chung paper: Fields Optionality and Measurability

I was reading the following paper: https://www.jstor.org/stable/2373011?seq=1#page_scan_tab_contents I had a question about a proof that is part of the paper. Here are some of the definitions and ...
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0answers
38 views

Prove that if $X$ and $Y$ are independent, then $h(X)$ and $g(Y)$ are independent in BASIC probability — can we use double integration?

In advanced probability we can just say: \begin{align} & P(h(X) \in A, g(Y) \in B) \\[6pt] = {} & P(X \in h^{-1}(A), Y \in g^{-1}(B)) \\[6pt] = {} & P(X \in h^{-1}(A)) P(Y \in g^{-1}(B)) \...
1
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1answer
29 views

Showing that $\{(x,x):x\in [0,1]\}$ is in $\cal{B}([0,1])\otimes\cal{B}([0,1])$

I am having a hard time showing that $\{(x,x):x\in [0,1]\}$ is in $\cal{B}([0,1])\otimes\cal{B}([0,1])$. ${\cal{B}}([0,1]) = \{[0,1]\cap B: B \in \cal{B}(\mathbb{R})\}$. Any hints would be ...
10
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1answer
66 views

Non-invertible measure preserving transformations of $\mathbb{R}^n$

I am looking for particular examples of measure-preserving transformations of $\mathbb{R}^n$ (with Lebesgue measure) to get a better idea of how they behave. A large family of such transformations ...
5
votes
2answers
139 views

Showing the set of real values for which the pre-image has measure greater than zero is measure zero

The question is stated as follows: Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is measurable, then the set $E = \{x \in \mathbb{R} \ | \ m(f^{-1}(x)) > 0 \}$ has measure zero. This ...
2
votes
1answer
73 views

$\int_X f^p d\mu = p\int_{[0,+\infty)} t^{p-1}\mu(\{x\in X: f(x)>t\}) d\mu_t$ for any natural $p\ge 1$

Let $f:X\mapsto[0,+\infty)$ be a non-negative measurable function defined on the space $X$, endowed with the complete $\sigma$-additive, $\sigma$-finite, measure $\mu$ defined on the $\sigma$-algebra ...
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1answer
19 views

Can any positive measure set in a standard Borel space be approximated in measure by an open set it contains?

Standard Borel probability spaces (a Polish space with a measure on it's Borel sigma algebra) satisfy the following: (*) For any Borel set $E$ of positive measure and any $\epsilon>0$, there is $F\...
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2answers
40 views

Are the following two integrals equivalent?

I am new to Royden's Real Analysis and am wondering what the relationship between Lebesgue and Riemann integrals. For example, are the two integrals $\int_{\mathbb R} 1_{[0,1]}dm$ and $\int _0^1 1 ...
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1answer
43 views

Is there a way to write conditional expectation as an integral?

Let $E[X|G]$ be a random variable that is G-measurable and satisfies the partial averaging property, then we know that $E[X|G]$ is a conditional expectation. This is the definition I saw from Shreve's ...
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1answer
28 views

Showing that $\{x:u(x)\le v(x)\}$ is measurable where $u,v$ are measurable

Let $u,v$ be measurable functions from a measurable space $(X,\cal{A})$ to $(\mathbb{\bar{R}},\cal{B}(\mathbb{\bar{R}}))$. How can I prove that $\{x:u(x)\le v(x)\}$ is in $\cal{A}$? It is of course ...
1
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1answer
29 views

Modified cantor set can be identified with binary sequences

I'm reading this paper on sets of uniqueness, and on page 15 the author constructs the following Cantor-like set: The Cantor set in the interval $[0, 2π]$ is constructed by removing the middle $...
0
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1answer
36 views

Suppose a sequence's subsequences have at least one subsubsequence that converges almost surely to $X$. Prove convergence in probability

Probability with Martingales What I tried: 'only if' Suppose a sequence converges in probability to $X$. By $d$ there exists a subsequence that converges almost surelyto $X$. Then by $a$, ...
0
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1answer
56 views

Does $\mathbb E[X] < \infty$ imply that $\mathbb E[e^X] < \infty$? [closed]

Let $X$ be a non-negative random variable with finite expectation. Is this sufficient to say that $\mathbb E[e^X] < \infty$? Jensen's equality clearly shows that if $\mathbb E[X] = \infty$ then ...
1
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1answer
39 views

Conditional expectation and independence on $\sigma$-algebras and events

In many statistics papers, proofs might proceed as follows: Under the event $A$, the random variables $X$ and $Y$ are independent. (Often this means that on $A^C$, they might be dependent). Then some ...
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1answer
34 views

Direct image of a sigma algebra

Let $(X,\cal{A}$) and $(X',\cal{A}')$ be two measurable spaces and $T:X\to X'$ a bijective measurable map. The claim is that $T(\cal{A})$ is a sigma algebra on $X' \iff T^{-1}:X'\to X$ is measurable....
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2answers
44 views

The Jordan Decomposition Theorem, Folland

The Jordan Decomposition Theorem - If $\nu$ is a signed measure, there exists unique positive measures $\nu^+$ and $\nu^-$ such that $\nu = \nu^+ - \nu^-$ and $\nu^+\perp \nu^-$. Attempted proof - ...
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1answer
36 views

$\lim_{p \to \infty} \|f\|_p = \|f\|_{\infty}$: is convergence monotone when $\mu(X) \leq 1$?

This question is related to Exercise 3.3.7(b) in Cohn, Measure Theory, 2nd edition, which reads as follows: Let $(X, \mathcal A, \mu)$ be a finite measure space, and let $f$ be an $\mathcal A$-...
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1answer
25 views

Proving the Hahn Decompostion Theorem from Folland

The Hahn Decomposition Theorem - If $\nu$ is a signed measure on $(X,M)$, there exists a positive set $P$ and a negative set $N$ for $\nu$ such that $P\cup N = X$ and $P\cap N = \emptyset$. If $P',N'$ ...
2
votes
1answer
47 views

For which $p$ is $\frac{1}{x^a+x^b}$ in $\cal{L}^p$?

Let $f(x)=\frac{1}{x^a+x^b}$ with $x,a,b>0$. For which $p\ge1$ is $f$ in $\cal{L}^p(\lambda)$ over the interval $(0,\infty)$? Here $\lambda$ is the one dimensional Lebesgue measure. Attempt: We ...
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1answer
20 views

Showing left continuity of this distribution function

Suppose $(X,\cal{A},\mu)$ is a $\sigma$-finite measure space and $u$ is a nonnegative measurable function. Then the function $F:\mathbb{R} \to \mathbb{\bar{R}}$ given by $t\mapsto \mu(\{x: u\ge t\})$ ...
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2answers
72 views

What is the diagonal principle?

I'm guessing I'm supposed to show that there exists a subsequence that converges in probability quickly (hypothesis of $(c)$). What is the diagonal principle? Is that related to Cantor's ...
2
votes
1answer
25 views

Continuity of a characteristic function of a translated set

Let $E \subseteq \mathbb{R}$ be a measurable set. Is it true that $\chi_{E+t}(x) \rightarrow \chi_{E}(x)$ as $t \rightarrow 0$, where $E+t = \{x+t \, | \, x \in E\}$ for each $t \in \mathbb{R}$, ...
1
vote
1answer
44 views

Lusin's Theorem, Modes of Convergence

Background Information: Theorem 1.18 - If $E\in M_{\mu}$, then \begin{align*} \mu(E) &= \inf\{\mu(U):E\subset U, U \ \text{open}\}\\ &=\sup\{\mu(K):E\subset K, K \ \text{compact}\}\end{...
4
votes
0answers
61 views

Show that $\int_{E}{F(x,t)}d\mu\otimes\lambda=\int_{X}{\int_{[\varphi_1(x),\varphi_2(x)]}{F(x,t)d\lambda(t)d\mu(x)}}$

Let $(X,\mathcal{F},\mu)$ be a $\sigma-$finite measure space. Let $\varphi_1,\varphi_2:X\to\mathbb{R}$ functions in $\mathcal{M}(X,\mathcal{F},\mathbb{R})$ such that $\varphi_1(x)\leq\varphi_2(x)$ for ...
1
vote
1answer
39 views

Halving a measureable set

Let $X$ be a set with a finite Lebesgue measure (e.g. a subset of the unit interval), and $u,v$ two measures such that: $$v(X) > u(X)$$ Does there exist a subset $Y\subset X$, with Lebesgue ...
14
votes
2answers
218 views

$f : \mathbb{R} \to \mathbb{R}$ (Lipschitz) continuous implies $f(A)$ is Borel for all Borel $A$.

Full question: Let $(\mathbb{R}, \mathfrak{M}, m)$ denote the measure space $\mathbb{R}$ equipped with the Borel $\sigma$-algebra and the Lebesgue measure. Suppose $f : \mathbb{R} \to \mathbb{R}$ is ...
3
votes
1answer
77 views

Construction of a measure space from some weird functional

Here is the complete problem I am trying to solve, but currently, I am just interested in proving that $\Sigma$ is a $\sigma$-algebra. Let $X$ be a set and $\phi: 2^X \to [0, \infty]$ be a ...