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

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Show a function is Lebesgue integrable

Hi I am struggling with a question but really I am struggling more with the concepts behind it so any help would be appreciated. Q/ Let $f(x)=x^{-\frac{1}{2}}$ for $x\in(0,1)$ and 0 otherwise. Let ...
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0answers
29 views

Integral equality

I have recently started studying function spaces (namely Lorentz endpoint spaces). Let $\varphi:[0; \infty)\rightarrow\mathbb{R}$ be a non-negative concave function such that ...
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1answer
34 views

Proving a function is Lebesgue integrable

I need to prove that $$\frac{|x|^\alpha}{1+x^2}$$ is Lebesgue integrable for $\alpha \in [0,1)$ but I'm not sure how to do this. I first tried expanding this using the Taylor expansion to show it is ...
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1answer
21 views

Help correct my misunderstanding regarding the measure of the unit ball in $\mathbb{R}^{n}$

I've confused myself over something elementary, so I'd appreciate it if someone can correct my misunderstanding. Consider the unit ball $B^{n}$ on $\mathbb{R}^{n}$. It is clear that $B^{n}$ does not ...
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50 views

Real Analysis versus Measure & Integration

I am looking at next semester's class schedule at my school, especially at a graduate course named Measure & Integration. Officially it is described as "... an introduction to the principles, ...
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1answer
31 views

Help understanding the equivalence of these two statements

Help understanding the equivalence of these two statements Let $\Omega $ and $S $ be sets and $Y : \Omega \mapsto S $ $\Sigma $ is a $\sigma $-algebra on $S $ $X: \Omega \mapsto \mathbb R $ Now I ...
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1answer
50 views

Is in every set of lebesgue measure $>0$ on $\mathbb R$ a “almost compact” set included

Let $\lambda$ be the Lebesgue measure on $\mathbb R$. Let $A \subset \mathbb R$ be a measurable set such that $\lambda(A)>0$. Question: Can we always find two measurable sets $K_1, K \subset ...
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1answer
55 views

Show that there exists two closed sets $A, B$ such that $m(A)=m(B)=0$ but $m(A+B)>0$.

I think this is a partial solution to this problem in Stein and Sharkarchi's Real Analysis. Since the most obvious choice for a closed set with measure 0 is the Cantor set, $\mathcal{C}$, we should ...
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1answer
21 views

Finding an specific counterexample of the interchanging of the limit and integral.

Hi I found the following exercise where I'm stuck, I'd appreciate any help. Thanks in advances Let $0\le f_n\to f$ and $\int f_n\to c>0$ pointwise. Show that $\int \lim f_n d\mu\in [0,c]$ and ...
3
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1answer
86 views

Proving that $\overline{\int f \, d\mu}=\int \bar{f} \, d\bar{\mu}$

I need to show that $\overline{\int f \, d\mu}=\int \bar{f} \, d\bar{\mu}$, where $\mu$ is a complex measure. The integral with respect to a complex measure is defined by $\int f \, d\mu = \int f ...
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0answers
25 views

Proving that Levy-measures are $\sigma$-finite

I have a question that pertains to Levy measures and, more specifically, I would like to show that they are $\sigma$-finite. They are not finite in general and it is easy to provide examples of this. ...
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1answer
49 views

Cardinality of sigma algebras.

Given $(\Omega,\sigma)$ a measurable space, it is not difficult to prove that if $$\forall w \in \Omega \hspace{0.4cm}\exists \hspace{0.1cm} C_w \in \sigma, w \in \cal{C}_w / \hspace{0.3cm} w\in D, ...
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1answer
24 views

Set is Lebesque measurable

I have following set $$M := \left\{ (x,y) \in \mathbb{R}^2 \mid \exists q \in \mathbb{Q}: \frac{q}{\pi} \leq \sin(xy)<\cos(y)+q^4\right\}$$ and are supposed to verify that the set $M$ is Lebesgue ...
3
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1answer
50 views

Weak convergence in $l_p$ implies pointwise convergence?

Could someone please share their thoughts on this one: Consider at $l_p(Y)$, for $1<p<\infty$ with the counting measure on $Y$. Show that if a sequence weakly converges in $l_p(Y)$ then it ...
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0answers
32 views

Counterexample in Probability Theory: non trivial property satisfied by indicators but not for all measurable functions.

When one has done or studied a few proofs in Probability Theory of results of the type 'The property P is satisfied by all measurable functions realizes that usually it is enough to prove that all ...
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41 views

Sentences about Lebesgue integrable function

In $\mathbb{R}$ with Lebesgue measure, we take $f\in L^1$ and we set $\hat{f}(t)=\int f(x) e^{ixt} dx$, for each $x$ $\ \ \ (i^2=-1)$ Show that: $\hat{f}$ is continuous $\lim_{t\rightarrow \pm ...
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Finding a general form of the density function when we have a four dimentional random variable.

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
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2answers
56 views

Prove that f:X→R is measurable if and only if for every a∈R, f−1(a,∞) ∈ M.

Suppose (X, M, μ) is a measure space. We say f : X → R is measurable if the inverse image of any Borel set in R is a measurable set in X: $S \in B_R ⇒f^{−1}(S) \in M$. Prove that f:X→R is measurable ...
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1answer
18 views

The relation of the measure of the boundary and the measure of the set

I am stuck with proving the following statement: Suppose that the set $A$ has measure $0$. I should prove that the boundary of A need not be of measure $0$. This is how far I could go. If the set A ...
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1answer
63 views

Convergence as for the norm [duplicate]

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $\|f_n\|_p \rightarrow \|f\|_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to ...
2
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1answer
23 views

Generalize the sentence in $R^d$

If $E \subset \mathbb{R}$ is Lebesgue measurable and $\phi(t)=m \left ((-\infty, t) \cap E\right )$, then $\phi$ is Lipschitz. How could we generalize this sentence in $\mathbb{R}^d$? If $E \subset ...
3
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2answers
67 views

Cardinality of non-Borel sets

Assume ZFC. Let $B\subseteq\mathbb R$ be a set that is not Borel-measurable. Clearly, $B$ must be uncountable, since countable sets are always Borel being a countable union of measurable singletons. ...
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1answer
29 views

Using Zorn's lemma to show a set is measurable.

If $S_j\subset \mathbb{R}^n$ and for all $j\in \mathbb{N}$ the set $S_j$ is measurable, show that $\bigcup S_j$ and $\bigcap S_j$ are measurable. I think I can use Zorn's lemma to show that this is ...
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Is there a $\sigma$-algebra which has countably many elements but not finitely many? [duplicate]

I couldn't think anything about this question from a book by Bass: Does there exist a $σ$-algebra which has countably many elements, but not finitely many? May I get a hint please? Thanks!
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27 views

Hausdorff dimension of closed intervals is not changed under $f(x)$

Let $f(x)=x^2$. Prove that for any $E\subseteq\mathbb{R}$ the dimension of image is not changed i.e $$\dim_HE=\dim_H(f(E))$$ Any set in $\mathbb{R}$ can be represnted as a countable union of ...
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1answer
17 views

Restriction of a finite measure to a set on unbounded function

So I have a measure $(X,\mathscr{F},\mu)$, possibly finite, or $\sigma$-finite, or a completely general finite measure. $B\in \mathscr{F}$ is a set of finite measure. For every measure set $A\in ...
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23 views

Exercise on Rudin about $R^2$ measurable

I'm thinking about exercise 9 on Rudin's Real and Complex Analysis chapter 8: $E$ is dense in $\mathbb{R}^1$ and $f$ is a real function on $\mathbb{R}^2$ such that: (a) $f_x$ is Lebesgue measurable ...
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$f$ square-summable on $X'\times X''$, $\varphi_m$ square-summable on $X'$ and $\int f\cdot\bar{\varphi}d\mu'$ square-summable on $X''$

Let $X:=X'\times X''$ be the product of measure spaces $(X',\mu')$ and $(X',\mu'')$, endowed with the Lebesge extension $\mu:=\mu'\otimes\mu''$ of product measure $\mu'\times \mu''$ defined by ...
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23 views

Is it necessary to take the inferior limit of this sequence of integrals

Suppose $A $ is a measurable set, and $\{h _n \} $ is a sequence of nonnegative simple functions such that $h _n \uparrow \chi _A $, where $\chi _A $ is the charachteristic function. I wonder why in ...
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1answer
13 views

Can I change the order of summation here?

Is it true that $\sum _{i = 1 } ^n \alpha _i \sum _{r=1 } ^{\infty } \mu(A _i \cap E _r ) = \sum _{r = 1 } ^{\infty } \sum _{i=1 } ^{n } \alpha _i\mu(A _i \cap E _r ) $ I know that I can change ...
2
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1answer
54 views

If a family $\{f_n\}$, is uniformly integrable, then also $\{|f_n|\}$ is.

I want to show that if a set of functions $\{f_n\}_n$ is uniformly integrable, then also $\{|f_n|\}_n$ is also uniformly integrable. How can I show this? My guess is to use separate $f_n$ into real ...
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1answer
21 views

Proving that sum of two measurable functions is measurable for conditional expectation

I'm trying to show something that seems pretty simple: $\mathbb{E}[aX + Y | \mathcal{G}] = a\mathbb{E}[X | \mathcal{G}] + \mathbb{E}[Y | \mathcal{G}]$ where the conditional expectation is defined ...
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22 views

How to apply Fubini's theorem in proof of Osgood's lemma

In the proof of Osgood's lemma for seperate holomorphicity, at one step we get that $$f(z)=\frac{1}{(2\pi \iota)^n}\int_{|w_i-\zeta_i|=r_i}\sum_{v_1,v_2,\ldots,v_n}\frac{f(\zeta)z_1^{v_1}\ldots ...
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1answer
15 views

Show the inclusions $\mathcal L^r(\lambda) \subseteq \mathcal L^s(\lambda)$ and $L^s(\lambda) \subseteq \mathcal L^r(\lambda)$ doesn't hold.

Suppose $0 < r < s < \infty$ and $\lambda$ is the Lebesque measure on $(\mathbb R, \mathcal B(\mathbb R))$. I want to show that $\mathcal L^r(\lambda) \subseteq \mathcal L^s(\lambda)$ and ...
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sigma-algebra for the class of measurable functions from a set to another set

What can be a sigma-algebra for the class of measurable functions from a set to another set ? I tried to use product sigma algebra but not succeeded.
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2answers
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Proving that f is measurable if g is measurable where $g(x, y) = |f(x) - f(y)|$

I need help understanding the proof of this question. Basically it involves Fubini's theorem and I can't seem to get a grasp on the proof (and I am pretty sure this question uses techniques from the ...
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1answer
21 views

Showing that a measure is lower continuous.

Here is the problem and my attempt. Let $\mu$ be a measure defined on sets $E_1, E_2, \ldots$ such that $E_{n+1} \subset E_n$ for all $n$. Additionally, let $\mu(E_1) \lt \infty$. Show that ...
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0answers
19 views

Is the following rule a stopping time in regards to reverse filtration?

Let $X_1, \dotsc, X_n \sim F$, where $F$ is a distribution function with support in $[0,1]$. For $t \in [0,1]$, define the sigma-algebra: $$ \mathscr{F}_t = \sigma(1_{\{X_i \leq s\}}\;,\; 1 \geq s ...
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1answer
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Extending measures counterexample

Given set X, measure $\mu$ and $\sigma$ algebra $\Sigma$. Say it is contained in a larger $\sigma$ algebra $\Sigma\subset \Sigma'$. Is there a way to extend the measure $\mu$ to $\Sigma'$? This must ...
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A basic measure theory question on Stochastic Process

Let $(Ω, F, P)$ be a probability space, $T$ some index set, and $(S, Σ)$ a measurable space. $X : T × Ω → S$ is a stochastic process, so it is measurable map. Let $S^T$ be the collection of all ...
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“Limit set” of infinite measure for a “Cauchy” sequence

Let $\{A_n\}$ be a sequence of sets $A_n\subset X$ of finite Lebesgue measure $\mu$ with the property that$$\forall\varepsilon>0\quad\exists N\in\mathbb{N}^+:\forall n,m\geq N\quad\mu(A_n\triangle ...
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1answer
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Show that there is a Lebesgue measurable subset with measure $a$

If $A\subset \mathbb{R}^d$ has positive Lebesgue measure, then for each $0\leq a<m(A)$ there is a Lebesgue measurable $E\subset A$ with $m(E)=a$. We have that $E\subset A \Rightarrow m(E)\leq ...
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1answer
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I want to show that $|f(x)|\le(Mf)(x)|$ at every Lebesgue point of $f$ if $f\in L^1(R^k)$

I want to show this, where $Mf$ is a maximal function, and I have attain $$Mf(x)-|f(x)|=\sup_{0\le r\le\infty}\frac{1}{B(x,r)}\int_{B(x,r)}(|f(y)|-|f(x)|)dm(y)$$ and I have no idea how to show that ...
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1answer
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On integration, measurability, almost everywhere concept

I have updated considerably, with my full solution. Feedback appreciated! Suppose $f$ is a nonnegative, measurable function and $\int f d\mu < \infty.$ Let $$h(w)=\begin{cases}f(w) \ \ \ \text{if} ...
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2answers
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What does $e^{\mu}$ mean for a measure $\mu$?

I have seen the notation $\int_M fe^{\mu}$ in some geometry books and I cannot even guess what $e^{\mu}$ might mean for a measure/form $\mu$ on the (symplectic) manifold $M$. Any clarifications are ...
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2answers
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Prove that every set that is content zero is also measure zero.

Prove that every set that is content zero is also measure zero. I understand that this is true, but am not sure how exactly to prove it.
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1answer
51 views

Why does $\sigma (X_t) \subset \sigma (X)$ hold?

We have a random process $X=\{X_t\;,t\in T\}$, where $X_t:(\Omega,\mathcal{A})\to(S_t,\mathcal{S}_t)$ are random variables. I am confused as to why does $$\sigma (X_t) \subset \sigma (X)$$ hold ...
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2answers
57 views

Lebesgue Integral: Convexity

Given a probability measure $\rho(\Omega)=1$. Consider a complex function $f\in\mathcal{L}(\rho)$. From the Riemann integral it is evident that: $$\int_\Omega f\mathrm{d}\rho\in\overline{\langle ...
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1answer
25 views

Complex Measures: Jacobian

Disclaimer This thread is meant to record results and written as jeopardy. ;) (For more details see: Answer own Question) Problem This is a follow-up to: Borel Measures: Jacobian Given a complex ...
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1answer
16 views

Borel Measures: Jacobian

Disclaimer This thread is meant to record results and written as jeopardy. :) (For more details see: Answer own Question) Problem Given a sigma-finite measure $\lambda$. Consider a finite measure ...