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

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Show that measure is regular

Let $\mathcal{B}$ denote the Borel-$\sigma$-algebra with respect to the Euclidean topology $\mathcal{T}$. Show that the measure $\lambda_{(0,1)}$ is regular. I start with outer ...
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1answer
34 views

Showing this function is $0$ a.e.

I would like to show the following: Suppose that $g \in L^1(\mathbb{R}^n)$ and $\int fg \,d \mu = 0$ for any $f \in C_0(\mathbb{R}^n)$. Then $g = 0$ $\mu$-a.e. I'm stumped on trying to find an ...
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1answer
32 views

Radon-Nikodym derivative for linear combination of measures

Given $(X, A, z)$ is a finite measure space, and for fixed $n$, $A_1, A_2,\ldots, A_n\in A$ such that $A_i\cap A_j = \emptyset$ for $i\neq j$. Let $a_1,a_2,\ldots, a_n\in R$ be distinct, and for $B\in ...
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1answer
31 views

Is the unit circle “stretchy” with respect to its norm?

Suppose we have a collection of metric spaces on $\mathbb{R}^n$, each of which has a different p-norm, $1\leq p \leq \infty$. ($p=2$ is Euclidean distance, $p=1$ is taxicab distance, etc.) Then, ...
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25 views

Simple functions are dense in L1

This is question 5.D in Bartle's Elements of Integration. If $f \in L(X,\mathcal X,\mu)$ and $\epsilon > 0$, then there exists a $\mathcal X$ − measurable simple function $\phi$ such that: ...
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16 views

Theorem 7.6 Bartle Elements of integration

Let $(f_n)$ be a sequence of measurable real-valued functions which is Cauchy in measure. Then there is a subsequence which converges almost everywhere and in measure to a mesurable real valued ...
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22 views

Second part pf the exercise 10.J of the elements of integration and Lebesgue of bartle's book

Let (X,X,$\mu$) be the measure space on the natural numbers X=$\mathbb{N}$ with the counting measure defined on all subsets of X=$\mathbb{N}$. Let (Y,Y,$\nu$) be an arbitrary measure space. A ...
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13 views

Approach a length by a BV norm

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $g: \overline{\Omega}\to \mathbb R^+$ defined by $g(x)=f(x)$ if $x\in \Omega$ and $g(x)=h(x)$ if $x\in \partial \Omega$, where ...
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0answers
16 views

Lower semi continuity for the norm of the speed

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $\gamma_n: [0,1]\to \overline{\Omega}$ be a sequence of Lipschitz functions which converges uniformly on $[0,1]$ to a Lipshitz ...
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4answers
120 views

Monotone Convergence Theorem for nonnegative functions (not quite a decreasing sequence)

This question suggests that the MCT for functions is to be applied, but I can't see how this could be done. Assume $g_n: X \to \bar{\mathbb{R}}$ is a sequence of nonnegative measurable functions ...
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22 views

Evaluate $\int_{0}^{\infty} \frac{\sinh bx}{\sinh ax} dx $

I need to evaluate the following integral $$\int_{0}^{\infty} \frac{\sinh bx}{\sinh ax} dx \space \space \space , \space \space 0<b<a$$ Here is my attempt - I can write $\sinh ax $ as ...
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0answers
8 views

Proposition in Daniel Kuhn et al paper “Primal and dual linear decision rules in stochastic and robust optimization”

Paper link:http://www.optimization-online.org/DB_FILE/2009/02/2218.pdf In this paper in page number 8 the autors make the following proposition: Let $\mathbb{P}$ be a probability measure on ...
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1answer
13 views

Every collection of measures on a compact space is uniformly tight

I am looking for a proper statement of the sentence in the title and its proof. First, let me give some context. I have a covariance stationary time series, $X$. The autocovariance function of $X$ is ...
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0answers
12 views

integration with delta function

Is there any way to calculate the following expression: $$\{\frac{\partial}{\partial t}\int|(1-t)p(x)+t\delta_{x_0}(x)-c|dx\}_{\text{at t=0}}.$$ Here, $p$ is a probability density function, ...
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1answer
43 views

Why $\lambda=0$ if $\lambda\perp \lambda$?

Rudin-RCA p.121 Let $(X,\mathfrak{M})$ be a measurable space. Let $\mu, \lambda$ be (complex) measures on $(X,\mathfrak{M})$ such that $\lambda \ll \mu$ and $\lambda \perp \mu$. Then, ...
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1answer
13 views

Continuous and discrete random variables defined on the same probability space?

I am confused on the definition of continuous/discrete random variables defined on the same probability space. Consider the random variables $X,Y$ defined on the same probability space $(\Omega, ...
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13 views

Properties of Kernel Integral inner Product of Gaussian Process

Can anyone give any reference / suggest how to get the rigorous mathematical properties of the following : $$ Y=\int_{a}^{b} K_{X} (t) \ f(t) \ dt $$ where $$f \sim GP (\mu(\cdot), R ...
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0answers
21 views

Does $f(x,y) = (x^2-y^2,xy)$ send measurable set to measurable set?

I know that $f(x,y) = (x^2-y^2,xy)$ is not lipschitz. Maybe it would be locally lipschitz, then I think apply standard argument that for each bounded measure zero set is measure zero, and extend it to ...
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1answer
59 views

Show that $\frac{\sin x}{x}$ is Riemann integrable on $[-1, 1]$

I need to show that the function $\frac{\sin x}{x}$ is Riemann integrable on $[-1 , 1]$. A function is called Riemann integrable if and only if it is bounded and continuous almost everywhere on its ...
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11 views

Bounding below Hausdorff measure of conected set

I'm trying to prove that for every connected set $E\subset\mathbb{R}$, $H^1(E)$, the Hausdorff measure is bounded below by $\text{diam}(E)$. In the answer there, it was suggested to use a Lipschitz ...
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1answer
11 views

Can we bound a Brenier map between uniform distributions with the Hausdorff distance between their supports?

Let $A,B$ be compact subsets of $\mathbb{R}^n$. Let $\mu_A$ (resp. $\mu_B$) be uniform probability measures over $A$ (resp. $B$). Then as a consequence of Brenier's theorem there is a one-to-one ...
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0answers
13 views

Volume Zero of Not Continuous Function

Show that a bounded real-valued function f on a closed interval $I$ of $E^n$ is integrable on $I$ if and only if the set of points of $I$ at which $f$ is not continuous is the union of a sequence of ...
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2answers
25 views

Proof outer measure satisfies monotonicity: $A \subseteq B \implies m^*(A) \leq m^*(B)$

Theorem: $$A \subseteq B \implies m^*(A) \leq m^*(B)$$ Proof Attempt: By definition, $m^*(B) = \inf\{\sum\limits_{k=1}^\infty |J_k||\{J_k\} \text{ is a cover of B }\}$, $m^*(A) = ...
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0answers
24 views

Given two finite measures $\mu$ and $\lambda$ on $R$, prove the following:

Given two finite measures µ and λ on R, show that (a) $\nu = λ + µ$ is a finite measure (b) $µ << \nu$ and $λ << \nu$ (c) Write $$\mu(E) = \int_E f d\nu$$ $$λ(E) = \int_E g d\nu$$ Let ...
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0answers
15 views

Question on bounded function and Radon Nikodym Derivative.

Given $\lambda << \mu$ finite measures, show that for any bounded function $g$ and $A \subset [0,1]$ $$\int_A gd\lambda = \int_A ghd\mu$$ where $h$ is the Radon-Nikodyn derivative ...
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0answers
21 views

Finding conditions on functions for absolute continuity of measure

Let $\mu$ and $\lambda$ be finite measures of $[0,1]$ which are absolutely continuous with respect to Lebesgue measure: $\lambda << m$ and $\mu << m$ $$\lambda(A) = \int_Af(x)dx$$ ...
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1answer
48 views

Does length $[0,1]$ = length $(0,1)$?

So we know that the length of an interval $[a,b]$ is simply $b-a$ but does this hold if the interval is open? Or if one of the sides are open, like $(a,b]$ or $[a,b)$? Also, can I just confirm that ...
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1answer
21 views

Continuous function differentiable almost everywhere, show f' is measurable

$f:\mathbb{R} \to \mathbb{R}$ continuous and differentiable almost everywhere, show that $f'$ is measurable. I feel like I should use some sequences, but I don't know how to start. Anyone?
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1answer
24 views

Monotonic increasing and convergence in measure

If for each $n\in\mathbb{N}$, $f_n$ is monotonic increasing on [0,1] and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ at every x at which f is continuous. I'm not sure whether this is right ...
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0answers
15 views

Relationship between Convergence in mean, convergence in measure and a.e. convergence

What is the relationship between convergence in mean under 1-norm (http://mathworld.wolfram.com/ConvergenceinMean.html), convergence in measure and a.e. convergence? I have shown that convergence in ...
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1answer
38 views

Using the limit definition to Prove that a Set has a Vitali Covering

Definition. For a real valued function $f$ and an interior point $x$ of its domain, the uppper derivative of $f$ at $x$ denoted by $\overline{D}f(x)$ is defined as follows: ...
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1answer
27 views

convergence of a subsequence of sequence converges in measure

If a sequence $(f_n)$ converge in measure to a function $f$, then every subsequence of $(f_n)$ converge in measure to $f$. Let $g_{n_k}$ a subsequence of $(f_n)$ then $|g_{n_k}(x)-f(x)|\leq ...
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0answers
19 views

Convergence in $L_p$

Let $f_n=n\chi_{[1/n,2/n]}$ in $\mathbb{R}$ with the Lebesgue measure defined on the Borel subsets of $\mathbb{R}$. Show that $f_n$ does not converge in $L_p$ to the $0$-function. $\left(\int (f_n)^p ...
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1answer
26 views

Measure of a set, convergence in measure

Let $(f_n)$ be a sequence of measurable real-valued functions which is Cauchy. Select a subsequence $(g_k)$ of $(f_n)$ such that the set $E_k=\{x\in X: |g_{k+1}-g_k(x)|\geq 2^{-k} \}$ is such that ...
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1answer
21 views

taking a limit inside a measure

dealing with the probability measure, I have that $$P(\cap_{n=1}^\infty A_n) = \lim_{M\to \infty}P(\cap_{n=1}^M A_n)$$ for some evens $A_n$. Could someone explain why we can do this?
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0answers
14 views

Linear transform of a strictly stationary time series

First, let me clarify what I mean by a strictly stationary time series. Let $(X_t)_{t\in \mathbb{Z}}$ be a sequence of random variables on some probability space. If it holds that $$(X_t, ...
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1answer
25 views

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg\, d\mu :\|g\|_\infty \leq 1\}$

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg \, d\mu :\|g\|_\infty \leq 1\}$ I know that Holders inequality implies $\int fg \, d\mu \leq ...
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1answer
17 views

Exercise 10.J of The elements of integration and Lebesgue measure Bartle's book

The part of the problem is the next. Let (X,X,$\mu$) be the measure space on the natural numbers X=$\mathbb{N}$ with the counting measure defined on all subsets of X=$\mathbb{N}$. Let (Y,Y,$\nu$) be ...
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1answer
20 views

Integral as a member of the closure of the convex hull of the integrand

Suppose that $X$ is compact and metric and let $g:X\to\mathbb R$ be a Borel map. Let $\mu$ be a Borel probability measure on $X$. Then it seems that $\int_Xgd\mu$ is a member of the closure of the ...
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2answers
43 views

Product measure problem

I wonder if you can help me out with this problem that I'm trying to understand (from an old exam, not homework): Let $(E,\mathcal{P}(E))\,$ (where $E= \lbrace 0,1 \rbrace )$ be a measurable space, ...
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1answer
93 views

Measure and set theory.

I have read that if we assume the continuum hypothesis then it can be proved or concluded tha there exist a set function μ that has the three following properties: μ(A) is defined for each set A of ...
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0answers
30 views

Linearity of Lebesgue measure

Suppose $\mu$ is the Lebesgue measure defined on $\Bbb R^k$, I want to show that $\mu$ has some kind of linearity, which seems intuitively correct: Suppose $A$ is a linear transformation on $\Bbb ...
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1answer
50 views

Hellinger integral properties - proof of equivalence for infinite product measures

I'm trying to prove that: Let $(\mu_k)_{k=1}^{\infty}$ and $(\nu_k)_{k=1}^{\infty}$ be sequences of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Consider the product measures on ...
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0answers
24 views

Wordy Measure Theory Books/Lecture notes

I found a set of notes http://www.gold-saucer.org/math/lebesgue/lebesgue.pdf for measure theory. In the preface it says "Style of exposition. We favor a style of writing, for both the main text and ...
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19 views

Integrability of a measurable function

Hi everyone: Suppose $(E_{n})$ is an increasing sequence of sets in $\mathbb{R}^{p}$ $(p\geq2)$ such that $\bigcup_{n}E_{n}= B$, a ball in $\mathbb{R}^{p}$. Suppose also that $f$ is a measurable ...
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0answers
54 views

Carother's “certainly” proof about measurable sets

Carother's Real Analysis text has the following Theorem. Can someone check if my proof is correct? $(i \Rightarrow ii)$ Let $E$ be a measurable. Let $I_k$ be open intervals, such that $$m^*(E) ...
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1answer
90 views

Why Are Some Sets Not Measurable?

I'm trying to understand why you can't evaluate a measure on generic sets (the ones in Banach-Tarski construction). That is, I want to know why when considering $m(X)$, we have to restrict our ...
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1answer
21 views

Definition of Hausdorff Measure: example question

I am studying the Hausdorff measure and dimension, but I am struggling to understand the reason that the $n$-dimensional Hausdorff measure is zero for a set with Hausdorff dimension $<n$. The ...
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29 views

Visualization of Fubini's Theorem

I understand that Fubini's Theorem is vital to evaluating double and triple integrals (via the equivalence of iterated integrals) especially in elementary multivariable calculus, and I know that it ...
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1answer
31 views

Fatou Lemma: Why is $\lim\inf f_n = 0$ where $f_n = \chi_{[n,n+1]}$

In this wildly popular post, there is a claim: I like to remember this by example; specifically let $f_n = \chi_{[n,n+1]}$. Then $\lim \inf f_n = 0$, and $\lim \inf \int f_n = 1$. So $f_n = ...