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

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6
votes
1answer
112 views

Generalization of Minkowski inequality

I am wondering if the following is true: Suppose continuous function $g: [0, \infty) \to [0, \infty)$ satisfying $g(0)=0$ is increasing and strictly convex and (therefore) invertible. Let $||f ...
2
votes
1answer
23 views

The irrational rotation is ergodic. The proof should use the idea of density point.

Consider $f_{\alpha}:S^{1}\rightarrow S^{1}$ the rotation of unit circle of angle $2\pi\alpha$, and let $\mu$ the Lebesgue measure in $S^{1}$. Let $\alpha$ irrational, show that $\left(f,\mu\right)$ ...
4
votes
1answer
50 views

approximation of measurable functions

Hi we know we can approximate measurable function by simple function however can we increase the conditions such that we can approximate by at most countable functions that is there exists sequence ...
29
votes
2answers
516 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
2
votes
1answer
28 views

Definition of $L^1(\mu)$, Lebesgue integrable function with respect to measure $\mu$

Here is an excerpt from Rudin's Real and Complex Analysis In Defintion 1.31, I am wondering why Rudin writes $f = u + iv$, where $u$ and $v$ are real measurable functions on X AND $f \in ...
0
votes
1answer
22 views

$\sigma$-algebra with cardinality $\aleph_0$ [duplicate]

Can a $\sigma$-algebra in a set $X$ have cardinality $\aleph_0$, the cardinality of the naturals? I do not have a clue on how to start with this? Can someone please give me a hint?
0
votes
0answers
14 views

definition for a function to be measurable

Hi I don't completely understand the following definition : Let A $\in$ F be nonempty , and let f : A $\rightarrow$ R denote a function. We will say that f is F/B*-measurable iff $f^{-1}(B)$ $\in$ F ...
2
votes
2answers
35 views

Lebesgue outer Measure of a face of rectangle in $\Bbb R^{n}$

Show that the outer measure of a face $I_1 \times \dots \times I_{i-1} \times \{a\} \times I_{i+1} \times \dots \times I_n$ of a rectangle $I_1 \times \dots \times I_n \subset \Bbb R^{n}$ is zero. ...
2
votes
2answers
50 views

An example where Egorov's theorem fails

This is p.62 of Folland Real Analysis book. Here the measure of X is supposed to be finite. But, I want to know the case in which the theorem doesn't work if X is of infinite measure. I tried to ...
1
vote
1answer
35 views

On the good set principle and sigma fields.

Following Probability and measure Theory by Ash (2000). let $\Omega$ be a set, let $C$ be a class of subsets of $\Omega$ and $A \subset \Omega$, we denote by $C \cap A$ the class $\{ B \cap A : B \in ...
1
vote
1answer
24 views

Differences in defining the packing (outer) measure

The definition of a packing measure in Falconer's Fractal geometry is given by I am assuming that $\mathcal{P}^s(F)$ as defined in 3.24 is an outer measure (this is not stated in the book). Now ...
0
votes
0answers
23 views

Proof of (part of) Dunford-Pettis theorem using ultrafilters

In P.A. Meyer Probability and Potentials, part of the proof for the Dunford-Pettis theorem is given in Page 20 (Theorem T23). I am looking at the proof of the following statement : Let $\mathcal{H} ...
0
votes
1answer
38 views

Example that the union of sigma algebra is not an algebra

I've tried to find the one, but failed to solve it. Some people asked similar question, but all the answers were about the case that "the union of sigma-algebra is not a 'sigma-algebra'". What I ...
1
vote
1answer
32 views

How can I solve the following exercise

How can I solve the following exercise : Why every simple function is measurable function ?? And The measurable function is simple function if its range is a finite subset of $R$
0
votes
0answers
19 views

Product $\sigma$-algebra on $\mathbb R^{\mathbb N}$

Let $\mathbb R^{\mathbb N}=\mathbb R\times\mathbb R\times\ldots$ be the space of all real sequences and endow it with product topology. Is the product $\sigma$-algebra generated by Borel subsets of ...
6
votes
2answers
95 views

If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e.

If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e. I let $E \subset \mathbb{R}^d$ be a finite measurable set. I try to break this into two cases: Case 1: If $f(x)=0$ ...
5
votes
2answers
319 views

What does Fatou's Lemma really say?

Fatou's lemma says that if $f_n:X \rightarrow [0,\infty]$ are measurable,then $$\liminf_{n\rightarrow \infty}\left(\int_X f_n \,\mathrm{d} \mu\right) \geq \int_X \liminf_{n\rightarrow \infty} f_n ...
1
vote
1answer
55 views

A function in $L^2([0,1])$ but not in $L^n([0,1] )$ for $n>2$ [closed]

Please indicate an example of a function in $L^2([0,1])$ but not in $L^n([0,1] )$ for any $n>2$?. The known examples are for $ n < 1 $ only.
1
vote
1answer
23 views

Dominated convergence and fundamental lemma of the calculus of variation

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= ...
0
votes
0answers
22 views

Fubini on non-$\sigma$-finite, non-complete measure spaces

Suppose $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ are measure spaces (not assumed to be $\sigma$-finite or complete). Define $\mathcal R$ to consist of all rectangles in $X \times Y$ of the ...
3
votes
0answers
47 views

Image of probability measure

Let $(\Omega,\Sigma,P)$ be a probability space. What is known about $P(\Sigma)$, the set of probabilities of events in $\Sigma$ by $P$? Clearly, $P(\Sigma)$ contains $0$ and $1$ since ...
1
vote
0answers
18 views

improvement of upper Lebesgue sum

In Pugh's real mathematical analysis, lower and upper Lebesgue sum are given as: $\underline{L}(f,Y)= \sum_\limits{i=1}^{\infty}y_{i-1}\cdot mX_{i-1}$ $\overline{L}(f,Y)= ...
0
votes
1answer
36 views

How can I prove like this

Let $(X,S)$ be a measurable space, and $f : X \to[0, +\infty]$ be a nonnegative function ($f \ge 0$). Then the function $f$ is measurable if and only is if there is an increasing sequence of simple ...
1
vote
0answers
30 views

Proof of the optional sampling theorem

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a Filtration on $(\Omega,\mathcal{A})$ $X=(X_n)_{n\in\mathbb{N}_0}$ be a nonnegative $\mathbb{F}$-supermartingale ...
1
vote
3answers
74 views

Prove that if $A$ and $B$ are measurable, then $\lambda(A)+\lambda(B)=\lambda(A \cup B)+\lambda(A \cap B)$

Prove that if $A$ and $B$ are measurable, then $\lambda(A)+\lambda(B)=\lambda(A \cup B)+\lambda(A \cap B)$ I tried to prove it using $A\cup B =(A\cap B^c)\cup (A\cap B)\cup(A^c\cap B)$ but ...
4
votes
2answers
69 views

A question about non-negative measurable function

Here is the proposition that I see in Rudin's Real and Complex analysis Suppose $f$ is a measurable function into $[0,\infty]$ and $E$ is a measurable set with respect to the measure $\mu$ Let $c$ ...
1
vote
0answers
45 views

The Lebesgue measure of set of small values of the Sinc function

Let the function $f\left( x \right)=\frac{\sin x}{x}$ for $x\ne 0$ and $f\left( 0 \right)=1$. Define $${{A}_{M,\varepsilon }}=\left\{ x\in \left[ -M,M \right]:\left| f\left( x \right) \right|\le ...
2
votes
1answer
35 views

If $\mu$ is $\sigma$ finite and $f_n \rightarrow f$ a.e then $f_n \rightarrow f$ uniformly on each $E_j$

If $\mu$ is $\sigma$ finite and $f_n \rightarrow f$ a.e, there exists $E_1,E_2, \ldots \subset X$ such that $\mu((\bigcup_{1}^{\infty}E_j)^{c})=0$ and $f_n \rightarrow f$ uniformly on each $E_j$ My ...
1
vote
1answer
48 views

A question about measurable functions

This is an exercise problem I am stuck at. I think I have to construct the function f based on X and T, starting from X being a nonnegative simple function However, I can't find a reasonable way to ...
0
votes
1answer
20 views

Borel mapping, simple functions and step functions

Say in $R$, I know that every step function is a Borel mapping, but is it true that every simple function is a Borel mapping as well? I think my question basically reduces to the existence of a non ...
0
votes
1answer
35 views

Showing that functions of bounded variation are not closed under composition

Find functions $g: [a, b] \to [c, d]$ and $f: [c, d] \to \mathbb{R}$ both of bounded variation, with f continuous, so that $f \circ g$ is not of bounded variation. This occurs as a request for a ...
0
votes
0answers
20 views

Question about criterion of measurability of a function

I am reading Rudin's Real and Complex Analysis, and in Theorem 1.12, Rudin states that Let $M$ be a $\sigma$-algerba in $X$ and $Y$ be a topological space. Let $f$ be a function from $X$ into $Y$ ...
1
vote
1answer
25 views

Martingale property on a $\sigma$-algebra generated by intervals

Consider the unit interval $I=[0,1]$ equipped with the Borel-$\sigma$-algebra on $[0,1]$ and the Lesbesgue measure $\lambda$. Now let $f$ be an integrable function on $I$ and define for ...
0
votes
1answer
32 views

Prove that an absolutely continuous cdf is continuous

Let $F(x_1,\ldots,x_d)$ be an absolutely continuous distribution function. How to prove that $F$ is continuous? Thank you.
0
votes
0answers
21 views

measurable selection for almost-minimizers of an irregular functional

I'm faced with the following problem: I have a functional $F$ defined on $H^1$ curves $[0,1] \rightarrow \Omega \subset \mathbb{R}^n$ where $\Omega$ is either a compact subset or the whole ...
1
vote
0answers
73 views

Positivity of density for sum of dependent random variables

Let $\{\xi_i\}_{i\geq 0}$ be a sequence of iid random variables that are uniform on a d-dimensional box $B_1(0) = [-1,1]^d$. Let $\{A_i\}:\mathbb{R}^d \to \mathbb{R}^d$ be invertible matrices with ...
0
votes
1answer
29 views

measurability condition

A set $A \subset X$ is $\mu$ measurable if \begin{equation} \mu(E)=\mu(E \cap A)+ \mu(E \setminus A) \text{ for all $E \subset X$} \end{equation} Does this work equally well for $\mu$ an ...
1
vote
1answer
17 views

M-measurability of limit function : $f_s$ when $s\in \mathbb{R}$

Suppose that for any $s\in \mathbb{R}$ there is given an $M$-measurable function $f_s : X \rightarrow [-\infty, \infty]$. Suppose that $\lim_{s \to\infty} f_s(x)$ exists for all $x \in X$. Prove that ...
1
vote
1answer
22 views

Proving $\chi_{f^{-1}(A)}(i) =(\chi_{A} \circ f)(i)$

Does this work for any $f$? \begin{equation} \chi_{f^{-1}(A)}(i) = \begin{cases} 1, & \text{if $i \in f^{-1}(A)$} \\ 0, & \text{if $i \not\in f^{-1}(A)$} \end{cases} = \begin{cases} ...
0
votes
0answers
14 views

Maximal exponent of integrability

Let $(X,\mathcal B,m)$ be a finite measure space. It is well-known that $L^p(X,m)\subset L^q(X,m)$ for $q<p$ and that the inclusion may be proper. I wonder if there is a function in $f\in L^q$ ...
0
votes
0answers
37 views

Construct a closed set $F \subset$ [0,1] which contains only irrationals so that m(F)>0.

Question:Construct a closed set $F \subset$ [0,1] which contains only irrationals so that $m(F)>0$. I wanted to find an open set containing Q and has measure <1 but I failed. Any hint?
1
vote
0answers
23 views

Countably Generated Sigma field

I need to show that a sigma field $\mathcal{G}$ is countably generated if and only if there exists a random variable $X$ such that $\mathcal{G}=\sigma(X)$. For the sufficiency part, I would say ...
4
votes
1answer
40 views

Abstract Integration in Elementary Probability Theory

In measure theoretic probability I often see these two notations for the expectation of a random variable expressed as an abstract integral. $$ \int_\Omega X(\omega) \mathbb{dP(\omega)} = \int_\Omega ...
0
votes
0answers
26 views

How can I prove like this exercise [duplicate]

Let we have $X=R$ the set of real numbers and $B_R$ is borel algebra and $2^X=P(X)$ How can I prove that $B_R$ is not equal to $P(X)$
3
votes
0answers
38 views

Are two random variables with the same distribution related by an isomorphism of an extended probability space?

Let $(\Omega, {\cal B}, P)$ be a probability space and let $X$ and $Y$ be two random elements $\Omega \to T$ where $T$ is some measurable space. Assume that the distributions of $X$ and $Y$ are equal. ...
1
vote
1answer
42 views

If $Y\in\mathcal{L}^1$ and $(\mathcal{F}_t,t\in I)$ is a filtration, then $\text{E}[\text{E}[|Y|\mid\mathcal{F}_t]]=\text{E}[|Y|]$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $(\mathcal{F}_t,t\in I)$ be a filtration on $(\Omega,\mathcal{A})$ $Y\in\mathcal{L}^1(\operatorname{P})$ and ...
0
votes
0answers
27 views

If a sequence of uniformly integrable random variables converges in distribution, the sequence of corresponding expectation values converges too

Let Let $\operatorname{P}$ be a probability measure $(X_n)_{n\in\mathbb{N}}\subseteq\mathcal{L}^1(\operatorname{P})$ be uniformly integrable and converge in distribution to $X$ Since ...
1
vote
0answers
20 views

Weak Convergence in Metric Space proof

I have been reading Billingsleys book where I came across this theorem and proof. I am having difficulty understanding the theorem/proof. I feel there is a better, more complete way to prove it. Does ...
2
votes
2answers
27 views

Proof of a theorem saying that we can make a measurable function continuous by altering it by a set of arbitarily small measure

I found the following problem in the book of kolmogorov fomin Introductory real analysis (p.293 problem 10) which I have no idea how to show. Prove that a function $f$ defined on a closed interval ...
0
votes
1answer
37 views

If f is Lipschitz with constant K, then $|f(E)|_e \leq K |E|_e$ for EVERY subset E of the domain

Let $A \subseteq \mathbb{R}$ be measurable and $f: A \to \mathbb{R}$ be a Lipschitz function with constant K, i.e., for any x, y in A, $|f(x) - f(y)| \leq K|x - y|$. Prove that for any $E \subseteq ...