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

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question 3.29 from Folland Real Anyalysis

If $F$ in $NBV$ is real-valued, then show $u_F ^+=u_P$ and $u_F ^-=u_N$ where $P$ and $N$ are the positive and negative variations of F. (Use Exercise 3.28) Source: Folland Real analysis exercise ...
2
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1answer
33 views

question 3.40 from Folland Real Anyalysis

Let $F$ denote the Cantor function on $[0, 1]$ (see $§1.5$), and set $F(x)= 0$ for $x<0$ and $F(x)=1$ for $x>1$. Let ${[a_n, b_n]}$ be an enumeration of the closed subintervals of $[0,1]$ with ...
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0answers
29 views

Prove that S=$\cup_{\varepsilon \ge0} \cap_{n=1}^{\infty}\cup_{m=n}^{\infty}E_m(\varepsilon)$ [on hold]

Let {$f_n$}$_n$ and $f$ be real valued function defined on $\mathbb R $ for $\varepsilon$>0 and $m\in \mathbb N$, define $E_m(\varepsilon)=${$x\in \mathbb R : |f_m(x)-f(x)|\ge \varepsilon$}. And let ...
1
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0answers
26 views

proving that the graph of a function is of Jordan measure zero

Let $f$ be an integrable function from $B$ to $[0,\inf]$ where $B$ is a sphere in $\mathbb{R^n}$. Exercise: For $f$ and $B$, the graph $$ \Gamma=\{(x,f(x)):x\in B\} \subset \mathbb{R}^{n+1} $$ is of ...
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2answers
52 views

Show that for any $1\leq p<\infty$, the set $L^1\cap L^p$ is a dense subset of $L^p$

Show that for any $1\leq p<\infty$, the set $L^1\cap L^p$ is a dense subset of $L^p$. Let $f\in L^p-L^1$. We need to find a sequence $\{\phi_n\}_n$ in $L^1\cap L^p$ converging to $f$. And I ...
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2answers
22 views

Steiner symmetrization of Lebesgue measurable set

I'm reading a proof in "Evans / Gariepy: Measure theory and fine properties of functions" of the following statemant: Theorem: Let $A \subset \mathbb R^n$ be $\lambda^n$-measurable, $a \in \mathbb ...
1
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1answer
9 views

Meaning of sampling i.i.d rvs from (random) probability measure?

Quote from book: "Consider an arbitrary atomic probability measure $\Gamma$ on unit sphere. Let $(\sigma_{l})$ denote an i.i.d sample from $\Gamma$." I don't understand the second sentence. Does it ...
3
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1answer
47 views

How many points are needed to intersect all elements in a sequence of measurable sets

Suppose $(X,\mathcal B, \mu)$ is a probability space and $n\in\mathbb N$ is an arbitrary but fixed integer. Is it true that if $m\in\mathbb N$ and if $A_1,\ldots,A_m\in\mathcal B$ with ...
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1answer
25 views

Need help with convergence of functional sequences [closed]

a) Does the sequence of functions $f_n=-{1\over n} χ_{[0,n]}$ converges to $0$? Which condition from Fatous theorem isn't satisfied? b) Can we apply Lebesgue's Dominated Convergence Theorem on this ...
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1answer
25 views

What can you say about union of two non measurable set. They are measurable or not?

What can you say about union of two non measurable set. They are measurable or not? Is it necessarily true?Thinking about α-cantor set I wonder if the complement of a non measurable set is a non ...
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0answers
30 views

Discrete measures and discrete kernels

This question was also posted here. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence ...
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0answers
18 views

How to solve this? Please Help me. [closed]

How to show that the outer measure of singletone set is zero?
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1answer
17 views

Question in the proof of Caratheodory's theorem

First, let me recall Caratheodory's theorem: Theorem (Caratheodory). Let $\mathcal{M}:=\{E:E\subseteq X,\theta A=\theta(A\cap E)+\theta(A\cap E^c),\forall A\subseteq X\}$ be the class of measurable ...
1
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1answer
28 views

Kolmogorov 0-1 law

Initial question: $X_n$, $n \in\mathbb N$, are independent real-valued random variables. Let $S_n$ be defined, for each $n\in\mathbb N$, by the sum: $S_n = X_1+X_2+...+X_n$. Prove that either the ...
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17 views

A two-stage experiment where the first stage has two independent outcomes

If $P(Y_1\in \cdot|X_1, X_2) = P(Y_1\in \cdot|X_1)$ and if $P(Y_2\in \cdot|X_1, X_2) = P(Y_2\in \cdot|X_2)$ and if $X_1$ and $X_2$ are independent, are $Y_1$ and $Y_2$ independent given $X_1, X_2$, ...
1
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2answers
54 views

Use Dominated convergence theorem to show that $f(x):=\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^3}$ is differentiable

Let $$f(x):=\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^3},$$ how can we show that f is differentiable everywhere by using the Lebesgue dominated convergence theorem? I know this theorem as saying ...
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1answer
17 views

Prove convolution $f\ast g\in L^\infty(\mathbb{R})$

Let $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$ ($1\le p,q<\infty:\frac 1 p+\frac 1 q=1$). Prove that $L^\infty(\mathbb{R}) \ni f\ast g$ (the convolution of them) and also prove that $$\Vert ...
2
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1answer
22 views

limsup of measurable functions is not measurable?

Suppose $f_r(x)$ is measurable for any fixed $r>0$, I was wondering whether $\limsup_{r\to 0^+}f_r(x)$ is measurable. I know the limsup of sequence of measurable functions is measurable, and I ...
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0answers
21 views

Set of all probability measures with finite support

Let $X$ be an uncountable set endowed with the discrete topology. Let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, and consider the subset $A$ of $\mathcal{P}(X)$ consisting ...
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2answers
71 views

Determining a charge through subsets

A charge is a finitely additive set function $c: \mathcal{P}(\mathbb{N}) \to [0, 1]$ such that $c(\mathbb{N}) = 1$ and $c(\{n\}) = 0$ for every $n \in N$. Here $\mathcal{P}(\mathbb{N})$ is the set of ...
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1answer
37 views

Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
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0answers
20 views

Measure of the Brownian Motion

Let $B_t$ be a Brownian motion and $B(x,r)$ the ball of x with radius r. Define $\mu_t(A):=\int_0^t 1_A(B_s)ds$ for $A\in \mathcal{B}(\mathbb{R})$. Now I'm not sure if the following equations are ...
2
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2answers
38 views

show $\int f_kd\mu\leq C$ for $f_k\geq0$, $\int fd\mu\leq C$

Let $(\Omega, \mathcal A,\mu)$ be a measure space and $f_k\rightarrow f$ a.e., $f_k\geq0$ and $\int f d\mu\leq C$ for some $C>0$. How can you show $\int f_k d\mu\leq C$ ? My attempt: I thought ...
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1answer
49 views

Some subtle questions in measure theory

This is the definition of 'almost everywhere' in Folland Real Analysis. If $(X,\mathcal{M},\mu)$ is a measure space, a set $E \in \mathcal{M}$ such that $\mu(E) = 0$ is called a null set. By ...
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0answers
60 views

$f$ is Riemann integrable iif the set of discontinuous points of $f$ has Lebesgue measure zero

This is a well known result in mathematics, but it's my first time attempting to prove it. I'm following the second book of Analysis from Folland. Below are the notations used and the theorem, from ...
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2answers
19 views

Prove this function is lower semi-continuous

Let $X$ be a metric space, and $B$ his borel $\sigma$-algebra. Fix $r>0$ Let $\mu$ be a probability measure on $(X,B)$ and define $f(x)=\mu(B(x,r))$. Show that $f$ is lower semi continuous. What ...
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20 views

I've followed the proof in the book - it seems more difficult than it has to be (basic measure theory, more real analysis)

This basically involves summations, so I'm tagging it real analysis. If $A\subset B$ then $\mu(A)\le\mu(B)$ - take this as fact, and $\mu:S\rightarrow\mathbb{R}\cup\{+\infty\}$. So a function that ...
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0answers
30 views

Show that $S$ is a $\sigma$-algebra

Let $C$ be any class of subsets of a set $X$. Let $Y \subset X$. Let $A(C)$ be the algebra generated by $C$. Let $S:=\{E \cup(B \cap Y^{c})\mid E \in S(C \cap Y)$, $B \in S(C)$}. Show that $S$ is a ...
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1answer
30 views

borel-measurable function is pointwise limit of a sequence of continuous functions, wich is uniformly bounded

Let $H$ be a Hilbert space over $\mathbb{C}$, $A\in L(H)$ ( $A:H\to H$ is linear and continuous) and let $A$ be self-adjoint. Consider the spectrum of A, $\sigma(A)$ and $f:K\to \mathbb{K}$ a bounded, ...
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1answer
20 views

Show that $S(f^{-1}(C))=f^{-1}(S(C))$

Show that $S(f^{-1}(C))=f^{-1}(S(C))$, where $f:X \to Y$ is a function and $C$ is a non empty family of subsets of $Y$. And $f^{-1}(C):=\{f^{-1}(c)|c \in C\}$ and $S(C)$ is the $\sigma$-algebra ...
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1answer
8 views

Is the weak limit of a sequence of translation-invariant measures itself translation-invariant?

Let $(X,\mathcal{B})$ be a measure space and $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of translation-invariant measures on it that converges weakly to the measure $\nu$ on $(X,\mathcal{B})$. ...
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0answers
26 views

Is a function from cantor set to R is measurable?

Is a function f from E to R is measurable? when E is cantor set. We are only known that the measure of E is 0.
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0answers
36 views

Union of two non measurable set is non measurable? [closed]

If $A$, $B$ be two non measurable set then is union of $A$ and $B$ is non measurable set?
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1answer
12 views

Exercise of Measurable Functions

Let ($\mathbb{N}$, $\mathcal{P}(\mathbb{N})$) and $f(n)$= $\tfrac{(-1)^{n+1}}{n} $ with $n\in \mathbb{N}$, I need to prove that this function is measurable or not Could give me some suggestions
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1answer
17 views

Sequence of continuous functions convergent to $0$ with the integral equal to $1$

I am looking for a sequence of continuous functions $\{f_m\}$ defined in $A\subset\mathbb{R}$ with $\lim\limits_{m\to\infty} f_m=0$ such that $\int_A f_m \;d\mu=1$. The problem I have is with the ...
5
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1answer
38 views

Convolution of two indicator functions can't be constant

Let $A,B \subset S^1$ be measurable sets (considering $S^1$ with say the lebesgue measure). I'm trying to prove that if the convolution $1_A*1_B$ is constant then one of $A$ or $B$ is a full measure ...
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0answers
18 views

Fatou's Lemma and Counting Measure

I have a vague problem in a Measure and Integration book here. They ask me to consider $\mu$ the counting measure in $\mathbb{N}$ and interpret Fatou's lemma, monotone and dominated convergence ...
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1answer
27 views

Measure on compact

I am tryying to solve exercise: Let $K \subset R$ is compact with no isolated point. Find Borel nonatomic measure $\xi$ on $R$ such that $\operatorname{supp}(\xi)=K$ Thank's on all help.
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1answer
45 views

Convergence of $f_n(x) = 2^n \cdot F(2^n (x-a_n))$ with $F(x) = e^{-x^2}$ with different notions of convergence.

I had my measure theory exam this morning, and one exercise was the following: I really can't see a solution. During the semester, we talked about almost everywhere convergence, almost uniform ...
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0answers
20 views

Restating a $\ge$ constraint as a limit involving $>$

Restating a $\ge$ constraint as a limit involving $>$ I've countered this before, long ago, but I've tried and cannot remember what I did. I am trying to solve something of the form that goes: ...
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1answer
15 views

What can we say about the absolute continuity of the joint distribution of $(X,Y)$?

If the distribution of $X$ is absolutely continuous with respect to the counting measure, and distribution of $Y$ is absolutely continuous with respect to the Lebesgue measure, what can we say about ...
3
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1answer
31 views

Proof that something is a sigma-field

For the first inclusion I have : C = {$(H \cap A) \cup (H^c \cap B)$} $\subset$ $A \cup B$ $\subset$ $F \cup {H}$ $\subset$ $\sigma(F \cup${H}) For the first inclusion can I say A U B is subset ...
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18 views

Example where the disintegration theorem does NOT hold

There are various conditions which guarantee that the disintegration theorem will hold, i.e. a joint probability distribution over a product space may be decomposed as a marginal probability and a ...
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1answer
66 views

Question about sigma field

My question lies in the opposite inclusion Let $F$ denote any $\sigma$-field other than $2^{\Omega}$, and let H $\subset\Omega$ with the property that $H \not\in F$. Then $\sigma(F \cup \{H\})= ...
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15 views

Define $P(A)$ to be 1 or 0 according as there does or does not exist $e_A$ such that $A$ contains $(0.5,0.5+e_A]$ [closed]

On the field $\mathcal{B_o}$ in (0,1], define $P(A)$ to be 1 or 0 according as there does or does not exist $e_A$ such that $A$ contains $(0.5,0.5+e_A]$, $e_A>0$ depends on A. Show that $P$ is ...
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2answers
39 views

To prove a given set is a $\sigma$ algebra

I need to prove the following If $R$ is a $\sigma$ ring then $\{ E \subset X : E \subset R $ or $ E^c \subset R \}$ is a $\sigma$ algebra. Here now my claim is that $E_j \in R\ \forall i = ...
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0answers
15 views

Measurability preserving operators on $L^2$

Given $L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$, a $\sigma$-algebra $\mathcal{G} \leq \mathcal{F}$, a function $f \in L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$ which is $\mathcal{G}$-measurable, ...
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1answer
28 views

Show that $\mu$ is absolutely continuous w.r.t. $\mathcal{L}$ and find $\frac{d\mu} {d\mathcal{L}}$

Let $\mu$ be the unique Borel measure on $\mathbb{R}$ satisfying $\mu((a,b])=\arctan b-\arctan a$. Show that for any $\mu$-measurable subset $E$ of $\mathbb{R}$, $\mathcal{L}(E)=0$ implies ...
0
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1answer
7 views

For every $E \subseteq \mathbb{R}^d$ with $0 < |E|_e < \infty$ and $0 < a < 1$ there exists a cube Q such that $|E \cap Q|_e \geq a|Q|$

For every $E \subseteq \mathbb{R}^d$ with $0 < |E|_e < \infty$ and $0 < a < 1$ there exists a cube Q such that $|E \cap Q|_e \geq a|Q|$ Here the exterior Lebesgue measure $|E|_e$ is ...
8
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2answers
126 views
+50

Weak topologies and weak convergence - Looking for feedbacks

I am currently trying to get exactly what the weak and the weak* topologies are, in particular in connection to the concept of weak convergence in measure, however I am not completely sure on what I ...