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

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22 views

pointwise converges and integrals

Let $(N,P(N),\mu)$ be a measure space such that $\mu(A)=\sum_{n\in A}{1\over n^2} $ a. Let $ f_n = n^2 * 1_{\{n\}} $. Does the sequence converges pointwise? b. Find all functions $ f:N \rightarrow R ...
1
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1answer
28 views

Borel measure induced by the Cantor function?

In an example to measure being mutally singular, the book has an example I do not understand. First the book has the definition: Mutually Singular Measure Let $(\Omega,\mathcal{A})$ be a ...
0
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1answer
37 views

A problem from Real Analysis of Folland

I got stuck on this problem. For the first statement, I tried to use $\epsilon -\delta$ condition, but still couldn't come to conclusion. So can anyone please help me solve this or give me some clue ...
1
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1answer
30 views

Let $([0,1],\mathcal{B}([0,1]),\lambda)$, $\lambda$ Lebesgue measure in $[0,1]$.

Show that if $f$ is $p$-integrable then, for each $\epsilon>0$, exists a function $h$ which is continuous in $[0,1]$ s.t. $\|f-h\|_p\leq\epsilon$. Is there any simpler way to show it than ...
4
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2answers
56 views

If $\mu(|f_n|^p)$ is bounded and $f_n\to f$ in measure then $f_n\to f$ in $L^1$

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of real measurable functions s.t., (a) The sequence $\displaystyle(\int |f_n|^p\ \mathsf d\mu)_{n\in\Bbb{N}}$ is bounded. (b) The sequence ...
2
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2answers
63 views

$\mu(A \cap I) \le a \mu(I)$ implies $\mu(A) = 0$?

Let $\mu$ be lebesgue measure on $\mathbb{R}$, $0<a<1$. If $\mu(A \cap I) \le a \mu(I)$ holds for any interval $I$, can I say $\mu(A)=0$? I tried to construct a counterexample by considering ...
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2answers
57 views

Show that a function defined by an integral is differentiable

Define $$g(a)=\int_{0}^{\infty}\frac{\sin(ax)}{x}e^{-x}dx,\ \ \ \ \ \ a\in\mathbb{R}$$ a) Show that $g(a)$ is differentiable and compute $g'(a)$. b) Use this to compute $g(a)$. I have tried various ...
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1answer
24 views

Let $(X,\mathcal{F},\mu)$ be a measure space and let $g\in L^1((X,\mathcal{F},\mu))$.

Let $\phi:[0,1]\to\mathbb{R}$ defined by $$\displaystyle \phi(t)=\int_X \frac{t^3g}{1+t^2g^2}\ \mathsf d\mu$$ Show that $\operatorname{Im}(\phi)\subset\mathbb{R}$ and that $\phi$ is continuous. ...
0
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0answers
14 views

Query about estimating an integral in Heat Equation

While studying the Heat Equation (P-309) from the book : 'Front Tracking From Conservation Laws' by Holden & Risebro; I have gone through the following calculation: " $\int_{\mathbb R} ...
1
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1answer
19 views

Proof that the set of integrable real-valued functions is a vector space

From Folland's Real Analaysis: Modern Techniques and Applications: Proposition: Let $(X,\mathcal{M},\mu)$ be a fixed measure space. The set of integrable real-valued functions on $X$ is a real ...
2
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1answer
70 views

Question about notion $d\mu = fdv$ in Real Analysis of Folland

I'm reading the book Real Analysis of Folland, chapter 3 about signed measure, and there's some notion that confused me. In this book, he defines that $dv = fd\mu$ if $v(E) = \int_E{fd\mu}$, and ...
0
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0answers
45 views

Suppose $f $ is absolutely continuous and that both $f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. [closed]

Problem: Suppose $f: \Bbb R \rightarrow \Bbb R$ is absolutely continuous on every interval $[a,b]$, and that both $f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. ...
1
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1answer
21 views

Integrability of Riesz potential

Given $f\in L^1(\mathbb{R}^3)$, define $$\phi(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|y-x|}\,dy.$$ I was able to show that $\phi$ exists for almost all $x$ (I used the Lebesgue differentiation theorem). ...
1
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2answers
56 views

Why can I use Fubini' theorem on this function?

I used the fact that $\displaystyle \int_0^\infty\int_0^1 e^{-y}\sin(2xy)\,dxdy=\int_0^1\int_0^\infty e^{-y}\sin(2xy)\,dydx$ to solve $\displaystyle\int_0^\infty e^{-y}\frac{\sin^2(y)}{y}\,dy$. (The ...
4
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0answers
20 views

Haar measure on a profinite group is the inverse limit of the counting measure on its quotients?

I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to: Let $N_i$ be a basis normal subgroup neighborhoods of the identity in a ...
2
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1answer
42 views

The pointwise limit of a measurable function is still measurable?

This is a previous discussion but I just found I didn't get the answer I want... The question is as follows: Assume a sequence of Radon measure $\mu_n\to\mu$ in weak star sense. The domain of $\mu$ ...
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1answer
20 views

Measurability of a version of a random variable

If $X$ is a ($\mathcal{F}$-measurable) random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and $Y$ is a version of $X$ in the sense that $\mathbf{P}(X \ne Y) = 0$ and ...
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1answer
26 views

Compactness of signed measure?

This idea never comes to me but I just realize that I am making a serious mistake that the space of finite signed measure is weakly compact... We all know that the space of finite Radon measure is ...
2
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0answers
45 views

If equality of dual space of a Banach spaces implys the equality of pre-duals?

Assume $ X_1$ and $X_2$ are two Banach Spaces such that $X_1\subset X_2$, i.e., the element belongs to $X_1$ belongs to $X_2$. No assumption on norms. Then I would expect that the dual space of them ...
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2answers
17 views

Hausdorff dimension of graph of function

This question came up on an exam Decide the Hausdorff dimension of the graph of the following function for $x>0$ $$y = \log(1+x)\sin\frac{1}{x}$$ In the course, we only touched upon the subject ...
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1answer
65 views

Background for Graduate Real Analysis I Class

This semester, I have signed up for a graduate Real Analysis I course (really a course in measure theory/Hilbert Spaces/Lebesgue integration) and have thus far attended two lectures. However, from ...
2
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1answer
48 views

If $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$

Show that if $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$ Hint: Use Hölder's inequality. But I can't see where I should use it. I'm trying to use it in $\displaystyle\int |f|^p\,d\mu = ...
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1answer
21 views

Lusin property (N) for functions of several variables

I just read in a paper by Martio and Zeimer$^1$ that smooth functions ($C^1$) of several real variables have the have the Lusin property (N). I have two questions. First, could someone give me a ...
3
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1answer
33 views

If $f_n\to f$ in measure and $\mu(|f_n|^p)$ is bounded then $\mu(|f|^p)$ is finite

-> The sequence $(\int|f_n|^p\,d\mu)_{n \in \Bbb N}$ is bounded. -> $f_n\to f$ in measure. Prove that f is p-integrable. I'm trying to use the dominated convergence theorem. But I can't find an ...
3
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0answers
52 views

Is the completion of a measure space necessary?

Most important theorems in measure theory do not assume the completeness of measure spaces. Monotone convergence theorem, Dominated convergence theorem, and Fubini's theorem, to name a few. So I ...
0
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1answer
27 views

Conjecture about regular Borel measures and dense sets with no interior

Suppose that $(X,\tau)$ is a topological space and let $\mathscr B$ denote the Borel $\sigma$-algebra on it. Moreover, let $\mu:\mathscr B\to[0,\infty]$ be a regular Borel measure, that is, ...
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1answer
34 views

Help verify a solution showing $f\left(x \right)=\int_\Bbb{R} {{\chi _A}\left(y \right){\chi _B}\left( {x-y} \right)dy} $ is well-defined everywhere

The question is, Let $A,B⊂[0,1]$ be measurable sets with $|A|>1/2$,$|B|>1/2$ where $|*|$ denotes Lebesgue measure. Prove that a. $|A⋂(1-B)|>0$ where $1-B≔{1-x:x∈B}$ and conclude that ...
5
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2answers
91 views

Number of equivalence classes of functions of real variable with the a.e relation.

What is the cardinal of the set $\mathcal{F}(\mathbb{R};X)/ \sim$ where $\sim$ is the relation $f\sim g \iff \mu(\{x\in \mathbb{R};f(x)\ne g(x)\})=0$ and $|X|=|\mathbb{R}|$? I guess that is ...
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1answer
69 views

Show that $\int_0^\infty \frac{\sin(x)}{x}e^{-xt}\,\mathrm{d}x=\frac{\pi}{2}-\arctan(t)$; $t>0$

I did this Let $I=\int_0^\infty\frac{\sin(x)}{x}e^{-xt} \,dx$ Then, $\frac{\partial I}{\partial t}=\frac{\partial}{\partial t} ...
2
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1answer
41 views

Cumulative distribution function implication

How can I prove the following: Let $X$ and $Y$ be two random variables. Suppose that their cumulative distribution functions satisfies $F_X(x)=F_Y(x)$ for all $x$. How can I show that $X$ and $Y$ are ...
1
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1answer
28 views

Pointwise limit of a measurable function is still measurable, for weak star convergence measure

Suppose I have a sequence of functions $(f_n(x))$: $\mathbb R^N\to\mathbb R^M$ such that $|f_n(x)|=1$ a.e. $x\in \mu_n$ where $\mu_n$ is a finite Radon measure over $\mathbb R^N$, and $f_n(x)$ is ...
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1answer
26 views

Does the following result require the random variables to be independent?

I am sitting with the book Labelled Markov Processes by Prakash Panangaden, and on page 79 he defines what it means for a set of random variables on a probability space to be independent, and after ...
5
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1answer
35 views

Existence of approximating simple function

Let $(X,\mathcal F,\mu)$ be measurable space with $\mu(X)<\infty$. $\mathcal F$ is $\sigma$-algebra on X and $\mathcal F$ is generated by algebra $\mathcal F_0$. Prove that for every measurable ...
4
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1answer
42 views

if $\mu(X)$ is finite and $f$ is finite on X a.e then $\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0$

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. Prove that if function $f$ is measurable and finite on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$ I have been ...
0
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1answer
28 views

Proof that a homeomorphic image of a non-borel set is non-borel

This question seeks to expand the proof given in the answer to this question. I am weak in topology, and am wondering if someone can provide a proof of why a homeomorphic image of a non-borel set is ...
1
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1answer
37 views

Mean value theorem for Lebesgue integral

Let $f$ be a mesurable function and $g$ be integrable function, and $\alpha, \beta$ are real numbers such that $\alpha \leq f \leq \beta$ a.e . Prove that there exists a real number $\gamma \in ...
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0answers
43 views

$\displaystyle M_{\epsilon} (x):=\sup_{y:|y-x|<\epsilon}f(y)\ \textbf{ show that }\ M(x)=\lim_{\epsilon\to\infty}M_{\epsilon}(x)\,$ exists for all $x$ [closed]

Let $\,f\,$ be a bounded function on $\,\Bbb R^n$ and for $\,\epsilon >0,\,$ and let $\displaystyle \,M_{\epsilon}(x)=\sup_{y:\mid y-x \mid <\epsilon} f(y)$. Show that $\displaystyle ...
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1answer
36 views

Help with a question about $\liminf \limits_{n \to \infty } \int_X {{f_n}\,d\mu } $ and $\int_X {\liminf \limits_{n \to \infty } {f_n}\,d\mu } $

The question is Let $(X, \cal M, \mu)$ be a finite measure space and fix $E\in \cal M$. For each $n \in \Bbb{N}$ define the function $f_n:(X, \cal M) \to \Bbb{R}$ given by $$ {f_n}(x) = \left\{ ...
1
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0answers
28 views

Expectation in measure theory

I'm reading a book on measure-theoretic probability, and the author defines the expectation of a random variable $X$ on a probability space $(\Omega,\scr H,\mathbb{P})$ as $\int_\Omega Xd\mathbb{P}$, ...
8
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2answers
101 views

Show that a given set has full measure or measure 0

Let $E \subseteq \mathbb R$ be Lebesgue measurable. And $E + q = E$ for any rational number $q$. Show that either $E$ or its complement has measure $0$. I tried this problem for few hours but ...
0
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1answer
31 views

The weak star convergence of Jordan decomposition

Given $\mu_n$ and $\mu$ finite signed Radon measures on the domain $\Omega$. We assume $\mu_n\to \mu$ in weak* sense, i.e. $\int_{\Omega}\phi \,d\mu_n \to \int_{\Omega} \phi\, d\mu$ for all test ...
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3answers
46 views

Show $\forall \epsilon > 0$ there exists $\delta > 0$ such that $\int_E {|f|d\mu } < \varepsilon $ for all $E\in \cal M$ with $\mu(E) < \delta$

The problem is Let $(X,\cal M, \mu)$ be a measure space and consider $f\in L^1(X,\cal M, \mu)$. Show that for each $\epsilon > 0$ there exists $\delta > 0$ such that $\int_E {|f|d\mu } < ...
0
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0answers
17 views

limiting and monotonic decreasing double sequence of probability measures

I am trying to figure out the behavior of this double sequence of measures. If I have a probability measure $\mu_n$ which is indexed by $n$, and a set of intervals $\mathcal{I}_k$ indexed by $k$ with ...
0
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1answer
39 views

Outer Measure exercise

This comes from an exercise from Real Analysis by Folland. Let $\mathcal{A}\subset P(X)$ be an algebra, $\mathcal{A}_\sigma$ the collection of countable unions of sets in $\mathcal{A}$, and ...
3
votes
2answers
48 views

What's the relationship between a measure space and a metric space?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ ...
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1answer
33 views

If $f, g$ are measurable functions, then $f+g$ is measurable

Show that $f(x)+g(x)<a$ iff there exists rational number $r,q$ such that $r+q<a$ and $f(x)<r; g(x)<q$. Use this to prove if $f, g$ are measurable functions, then $f+g$ is ...
0
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2answers
31 views

if $\mu(X)$ is finite then $\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0$

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. a) Prove that if function $f$ is measurable on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$ b) Can we ...
0
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1answer
42 views

What does Lebesgue measure space look like?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ ...
1
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2answers
70 views

Null set squared is a null set

I'm attempting to find a solution to the following problem that doesn't involve splitting this into various cases. The question is: "If $m^*(E) =0$, show that $m^*(E^2) = 0$, where $E^2 = \{x^2 ...
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0answers
13 views

Do there exist equidistributed countable subgroups in (compact) Lie groups?

By an equidistributed countable subgroup I mean a countable subgroup (with a finite or possibly countable set of generators) that is dense in $G$ such that for any sufficiently nice function (Haar ...