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

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Intermediate Value Like Property for Lebesgue Measure

Below is a question from N.L. Carother's book Real Analysis. I've provided my attempt at a solutions, however, any feed back would be very appreciated. Suppose $E$ is a measurable subset of $\...
2
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1answer
39 views

Can anyone explain the connection between reverse fatou's lemma and Fatou's lemma?

Here is the version of reverse Fatou's lemma I am looking at. $E_n$ is a sequence of events. $P(\limsup E_n) \geq \limsup P(E_n)$ Here is Fatou's lemma. Let $f_1,f_2,\ldots$ be a sequence ...
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1answer
18 views

Definitions of inner and outer regular measures

The definitions can be found here. I am trying to figure out why it's defined specifically this way. More precisely- Why compact sets are used to define inner regularity while open sets are used for ...
2
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1answer
42 views

Does the image of positive measure set under homeomorphism also have positive measure?

Say I have a homeomorphism $f:A\longrightarrow B$ between open subsets $A$ and $B$ of $\mathbb{R}^n$. If $S\subset A$ has positive Lebesgue measure, does $f(S)$ also have positive measure? If so, do ...
2
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1answer
46 views

Why the space of complex measures is Banach?

I've read the proof from here: Space of Complex Measures is Banach (proof?) and understood the part that proves that constructed limit is complex measure. But the first part is a bit unclear for me. I ...
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1answer
31 views

limit superior and law of large numbers [on hold]

I am wondering whether the following result is true: Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. real-valued random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ ...
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0answers
17 views

Characterization of Banach sublattice of L^1

Let $(X, \Sigma, \mu)$ be a measure space and let $F\subset L^1(X,\Sigma,\mu)$ be a Banach sublattice of $L^1$ with the following properties: (1) If $f\in F$, $f$ real-valued, then $f\land 1\in F$ (...
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1answer
44 views

Characterization of measures such that $\frac{1}{x} \in L^1(H)$

Let $H$ be a finite measure on $(0,1)$. What conditions must $H$ fulfill, such that \begin{equation*} \frac{1}{x} \in L^1(H),\ \ \ \frac{1}{1 - x} \in L^1(H) \end{equation*} I'm trying to characterize ...
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0answers
17 views

Alternative definition of Lebesgue measurable set.

Consider $\mathbb{R^d}$ with Lebesgue measure $\mu$. Suppose that for any $\epsilon >0$, there exists a Lebesgue measurable set $F$ such $\mu^* (E\Delta F) < \epsilon.$ Then $E$ is Lebesgue ...
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38 views

How do I prove that $L^1$ functions are almost continuous?

Let $f\in L^1(\mathbb{R}^n)$ and let $\epsilon >0$. How do I prove that there exists a measurable $A\subset \mathbb{R}^n$ such that $m(\mathbb{R}^n \setminus A)<\epsilon$ and $f\upharpoonright (\...
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1answer
60 views

Radon Nikodym Thm: extending to $\sigma$-finite case

I am reading Bartle's "Elements of Integration". Radon-Nikodym Thm: Let $\lambda,\mu$ be $\sigma$-finite measures on a measurable space $(X,\textbf{X})$ and say $\lambda \ll \mu$. Then $\exists$ ...
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0answers
49 views

How can integrals be developed? [closed]

I wonder if it would be possible to develop integrals without any use of simple functions and "area approaches". Thank you.
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0answers
22 views

Sets not in a sigma-algebra

I have a question concerning some sets which are not in a given sigma-algebra. More precisely, I have two questions closely related: Let $\mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d), d \ge 1$, be ...
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0answers
33 views

Problem in measure theory. Which of the following are true? [closed]

Let $f : [a, b] \to \mathbb{R}$ be a measurable function. Then a) $ \ $ if $\int\limits_c^d f(x)dx = 0$ for all $ a \leq c < d \leq b,$ then $f = 0$ a.e. b) $ \ $ if $\int\limits_a^c f(x)dx = 0$...
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0answers
29 views

Well-definedness of Fourier transform of $f\in L^1(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ where the measure is taken to be the Lebesgue measure. The Fourier transform of $f$ is the function $\hat{f}$ defined as $$\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{...
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3answers
79 views

Real Analysis Folland, Proposition 2.6 Measurable functions

Question: Proposition 2.6 - If $f,g: X\rightarrow \mathbb{C}$ are $M$-measurable, then so are $f + g$ and $fg$. Attempted proof/brainstorm - Suppose $f,g: X\rightarrow \mathbb{C}$ are $M$-...
2
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0answers
26 views

Show linearity of a functional if it holds for nonnegatives

Consider a functional $G^+:L_p \to \mathbb{R}$. Here $L_p = L_p (X,\textbf{X}, \mu)$ is the collection of all integrable fns (f s.t. $\int \vert f \vert^p d \mu < \infty$ on the measure space $(X,\...
2
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2answers
38 views

Finite Borel measure on a compact Metric Space

Suppose that $\mu$ is a finite Borel measure on a compact metric space $X$ and that $\mu(\{x\}) = 0$ for all $x\in X$. Show that for every $\epsilon > 0$ there is a $\delta > 0$ such that for ...
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0answers
16 views

A question regarding the proof of Riesz Representation Theorem for the dual of Lp.

The theorem is stated below. Pasted following is the proof. Can anyone explain why $g \rightarrow S(g)-\int_X fgd\mu \ for\ all\ g\in L^p$ is continuous? And right above equation 15, why is $|g_n -...
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14 views

Can anyone explain one step in the proof of Helley's Theorem?

The theorem and its proof is pasted at the end of the post. Can anyone why this limit holds? What is a good way to think of it intuitively? $$lim_{k \rightarrow \infty} T_{n_{k}}(f_j) \ for\ all\ j?$$...
2
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1answer
31 views

Random set of rationals topological properties

Flip a coin (probability of heads is p, strictly greater than 0 and strictly less than 1) for every rational number. For each toss, if heads include the number in a set S, if tails exclude it. What is ...
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1answer
41 views

Capacity of a set in $\mathbb{R}^n$

The $2$-capacity of a set $\Omega$ sitting inside an open set $V \subset \mathbb{R}^n$ is given by $$\text{cap}_2(\Omega, V) = \inf_{u \in C^\infty_0(V), u|_\Omega \equiv 1} \int_V |\nabla u|^2 dx.$$ ...
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1answer
97 views

Infimum of lower semicontinuous functions

The following proposition is from the book Nicolae Dinculeanu Integration on Locally Compact Spaces: Let $H$ and $K$ be two compact Hausdorff spaces and $\alpha$ a continuous mapping of $H$ onto $K$. ...
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0answers
65 views

What does it mean for a function to be invertible 'almost everywhere'

It seems to me that the correct definition of a measure-theoretic inverse for a function f is a function g such that $f \circ g$ and $f \circ g$ are the identity almost everywhere. The problem I have ...
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0answers
61 views

Integral over Julia Set (Is my math correct?)

So I was answering this question about whether or not the Julia Set was self-similar in a known way. Of course it is, and that got me thinking. Even though the self similarity is nonlinear, what if ...
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1answer
27 views

Can anyone explain one step in the proof of Riesz Representation theorem?

I am trying to understand the following lemma from Royden's Real Analysis which is directly used to prove Riesz Representation theorem. The book in the proof states that " when $p = 1$. We must ...
1
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3answers
66 views

Real Analysis, Folland Problem 1.5.30 Borel measures

If $E\in L$ and $m(E) > 0$, for any $\alpha < 1$ there is an open interval $I$ such that $m(E\cap I) > \alpha m(I)$. Attempted proof/brainstorm - Let $E\in L$ with $m(E) > 0$ and suppose ...
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2answers
48 views

If function is measurable on an interval, is it measurable on its subinterval?

This is exercise 2.3 from "A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson: Let $[c,d]\subseteq[a,b]$. Show that if $f$ is measurable on $[a,b]$, then $f$ is ...
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1answer
28 views

Sequence of subsets $E_n$ of $[0, 1]$ with $m(E_n) = 1$ for every n but $m(\cap_{n=1}^\infty E_n) \neq 1$

This is actually a "prove or give counterexample" type of problem. The claim is that if any sequence of subsets $E_n$ of $[0, 1]$ have $m(E_n) = 1$ (m being the standard Lebesgue measure) for every n, ...
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1answer
48 views

Real Analysis, Folland Problem 1.5.29 Lebesgue measurable set

1.5.29 - Let $E$ be a Lebesgue measurable set. a.) If $E\subset N$ where $N$ is the nonmeasurable set described in section 1.1, then $m(E) = 0$. b.) If $m(E) > 0$, then $E$ contains a ...
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1answer
29 views

Real Analysis Folland Problem 1.5.28 Borel Measures

Problem 1.5.28 - Let $F$ be increasing and right continuous, and let $\mu_F$ be the associated measure. Then $\mu_F(\{a\}) = F(a) - F(a-)$, $\mu_F([a,b)) = F(b-) - F(a-)$,$\mu_F([a,b]) = F(b) - F(a-)$,...
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0answers
34 views

Center of a mass for linear $f$ over $E$ with $(x,y,z)\in E \iff (x\cos\theta-y\sin\theta,x\sin \theta+y\cos\theta,z)\in E$

Let $E\subset \Bbb{R}^3$ be a measurable set (i.e. $\int_{\Bbb{R}^n}1_{E}$ exists) and let $v(E)\ne 0$. Let $f$ be a linear function $f:\Bbb{R}^3\to \Bbb{R}$, and let $(x_0,y_0,z_0)$ be the center of ...
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0answers
32 views

Weak convergence of finite measure preserving transformations

I am reading King's paper "The commutant is the weak closure of the powers, for rank-1 transformation" and I am not able to show that: (0.1) "If the $T_i$ are invertible measure preserving ...
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2answers
27 views

The Lebesgue measure of the set of horizontal lines through the points of a subset $A$ of $\mathbb{R}$ with $\lambda(A)=0$

Suppose $A$ is a subset of the real line with $\lambda(A)=0$ and $H=\{(x,y):x\in A\}$. What is a natural idea behind proving that $\lambda(H)=0$ ? In fact, I wish to prove that the collection of ...
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2answers
36 views

Bound on integrable nonnegative function $F$ given inequality with compactly supported continuous functions.

Full Question: Suppose that $F$ is a nonnegative function that is integrable on $\mathbb R$ and there is a constant $C$ such that $\int_\mathbb R Ff \leq C\int_\mathbb R f$ whenever $f$ is a ...
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0answers
30 views

Sufficient (and necessary?) conditions for normality of Gaussian process integral

Question Let $X(\cdot)$ be a Gaussian process on $\mathcal{J}=[a,b]\subseteq\bar{\mathbb{R}}$ (extended real line) with mean $\theta(\cdot)$ and covariance $\Sigma(\cdot,\cdot)$. My goal is to find ...
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0answers
22 views

What is a sequence of functions that converge weakly in Lp, but not strongly?

I am reading Royden’s real analysis. In his book, a sequence of functions in Lp converges weakly if every bounded linear functional in the dual space converges in R. Can anyone discuss the ...
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26 views

show that $m(W\cap[x,\infty))=0$ as $x\to-\infty$? [closed]

Say I have a set $W$ of finite Lebesgue measure $\alpha$ and an interval $[x,\infty)$. Then how would I show that $m(W\cap[x,\infty))=0$ as $x\to-\infty$?
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22 views

Show that $f$ defined on the interval $(a,b)$ is not differentiable for every point in $E$ with $m(E)=0$

Let $E$ have measure zero contained in the open interval $(a,b)$. In a previous problem I showed that there is a countable collection of open intervals, $\{(c_k,d_k)\}_k$, contained in $(a,b)$ for ...
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1answer
27 views

If a sequence converges in measure, are convergent subsequences of it all converge to the same limit?

Let $f_n:X\rightarrow \mathbb{C}$ be a sequence of measurable functions such that $f_n\rightarrow f$ in measure. Let $f_{n_k}$ be a subsequence of $f_n$ such that $f_{n_k}\rightarrow g$ pointwise a.e....
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1answer
24 views

Jacobian for $X = YX'$ where $X',Y,X$ are $n\times n$ matrices?

I'm trying to work through this example on the wiki for Haar measures, showing that $$ \mu(S) = \int_S \frac{1}{|\det(X)|^n}\,dX $$ is a left Haar measure for $\mathrm{GL}(n,\mathbb{R})\subseteq \...
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1answer
41 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
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1answer
23 views

Variation of Egoroff's theorem.

Let $(X,\mu)$ be a measure space with a (positive) measure. Let $\{f_n : X \to \mathbb{R} : n =1 ,2,...\}$ be a sequence of measurable functions satisfy the following properties : For each $n\in \...
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19 views

Change of Variables for Integral over a Fractal

We define the integral of a function $f(x,y)$ over a fractal $F$ to be, $$(1) \quad \int_F f(x,y) \ d\mu(x,y)$$ Where $\mu$ is the normalized Hausdorff measure. Expressed another way, we have, $$\...
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1answer
38 views

Baire measurable sets

I got the following setting: Let $X$ a uncountable set, $p \notin X$ and $X^* = X \cup \{p\}$. Let $$\mathcal O := \{O \subseteq X^* \mid O \subseteq X \text{ or } p \in O \text{ and } X \setminus O \...
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1answer
26 views

Is there a meaningful measure on analytic functions?

Let $\mathcal{B}$ be the functions analytic on the unit disk and continuous on its boundary. With the supremum norm this becomes a Banach space. Is there any way to define a meaningful measure on ...
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0answers
16 views

Example of linear functional from Rudin's book

Rudin states that $\Lambda(f)=\int \limits_{X}fg d\mu$ is linear functional. We know that $\exists M$ such that $|g|\leqslant M$. Hence $|fg|\leqslant M|f|$ $\Rightarrow$ $\int \limits_{X}|fg|d\mu<\...
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1answer
20 views

Definition of complex vector space from Rudin RCA

This is definition of complex vector space from Rudin's book. He write that to each pair $(\alpha,x)$, where $x\in V$ and $\alpha$ is scalar there is associated a vector $\alpha x\in V$. That's right. ...
2
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1answer
60 views

Uniqueness of the uniform spherical distribution

Suppose that $X,Y$ are random vectors on some (possibly different) probability spaces mapping to $\mathbb R^n$ for some $n\in\mathbb N$. Suppose furthermore that $\|X\|=r>0$ for all realizations ...