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

learn more… | top users | synonyms (1)

1
vote
0answers
34 views

Prove that $f(x)=e^{-1 /\Vert x \Vert^\alpha}$ is measurable function.

Let $\Omega$ be a bounded domain in $\mathbb{R^N}$ and $f(x)=e^{-1 / \Vert x \Vert^\alpha}$. Prove that $f(x)$ is a measurable function. I wonder if I have to add more condition to $f(x)$. For ...
4
votes
1answer
70 views

Monotone increasing continuous function with $\int_a^b f' = f(b) - f(a)$ which is not absolutely continuous

If $f:[a, b] \to \mathbb{R}$ is continuous and real-valued, f' integrable on [a, b], and $\int_a^b f' = f(b) - f(a)$, must f be absolutely continuous? What if f is monotone increasing? For the ...
2
votes
1answer
41 views

Is this statement about measurable function true? [duplicate]

If $f$ is a measurable function on $[0,1]$, then there exists a measurable set $D\subset [0,1]$ such that $mD>0.9$, and a continuous function $g:[0,1]\to R$ such that $f=g$ on $D$. My question is ...
0
votes
3answers
46 views

$\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, $k < l$. [closed]

How do I show that $\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, with $k < l$?
1
vote
1answer
32 views

A class of functions dense in $L^2$

Suppose $f\in L^2([0,1],\Sigma,\mu)$. Is the class of all $$f=\sum_{i=1}^n \alpha_i (\chi_{A_i}-\chi_{[0,1]/A_i} )$$$A_i\in \Sigma$ to be dense in $L^2([0,1],\Sigma,\mu)$? Thanks.
2
votes
1answer
40 views

prove that $\int_{\Omega}|f_n-f_0|d\mu\rightarrow 0$ (By weaker assumption on Scheffé's lemma)

I'm dealing with this problem. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and ...
1
vote
0answers
40 views

Absolutely continuous iff continuous of bounded variation

I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ...
1
vote
0answers
23 views

Are $L1$ functions with a.e. finite support a.e. equal to a continuous function?

I was wondering about this: Let $f \in L^1(\Omega)$ and $\Omega\subset \mathbb{R}^n$ be compact, then $f$ is the $L^1$ limit of continuous functions with support in $\Omega$. Egorov's theorem tells us ...
1
vote
0answers
23 views

continuity at a point in a cadlag process

I am reading a proof that uses the fact: Let $(X_t)_{t \geq 0}$ be a cadlag process. We know that $X(\omega)$ has at most countably many discontinuities, for each $\omega \in \Omega$. It is then ...
2
votes
0answers
17 views

For $f\in L^1_{loc} (\Omega)$, $f=0$ almost everywhere in $\Omega$ provided $\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$

I need to show that $f=0$ almost everywhere in $\Omega$ provided $$\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$$ Here is how I have decided to proceed. Suppose there ...
1
vote
1answer
19 views

A problem about Markov kernel and the Monotone class theorem

Let $f: A\times B \times C \to R $ be a measurabled function, which is also bounded. And $p:B \times \sigma(C)\to R$ be a Markov kernel. Prove $g(x,y)=\int f(x,y,z)dp_y (z)$ is measurable with ...
1
vote
1answer
22 views

A relation between convergence in measure and pointwise convergence

Let $\{f_n\}$ be a sequence of measurable functions on $R$ with Lesbegue measure and $f$ be a measurable function. I have to show that $\{f_n\}$ converges to $f$ in measure if and only if any ...
1
vote
0answers
34 views

A measure on the space of probability measures

I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric ...
1
vote
0answers
27 views

Regularity of Special Measures

(1) Show that the counting measure on $\Bbb Z$ with the induced metric from $\Bbb R$ is regular. (2) Show that the delta measure with respect to a point $x_0$ on any metric space is regular. What I ...
1
vote
1answer
20 views

Proof of property of Lebesgue measurable subset of $\mathbb{R} $ with non-zero measure

I'm having trouble getting my head around how to prove the following: Let $A \subset \mathbb{R}$ with $m(A)>0$. Then $\forall \epsilon \in (0,1)\ \exists$ an open interval $I$ such that $m(A\cap ...
0
votes
1answer
28 views

$f_n$ converge to $f$ almost everywhere (measure theory)

If $\mu(\{w \in A: |f_n(w) - f(w)| > \epsilon\,\, \text{for infinitely many many $n$}\}) = 0$ for each $\epsilon > 0$, then $f_n \rightarrow f$ $\mu$-a.e. I can see how to solve this problem ...
0
votes
1answer
40 views

prove the equivalence between a null set and a limit

I'm asked to prove that for any non-negative, measurable and integrable function $f$ on $[0,1]$, we have $\lim\limits_{a\to 0}\int_{0}^{a}fdx=0$. I want to use the theorem that for null set E, such a ...
0
votes
2answers
34 views

Almost everywhere convergence for integral

I have a question about the following proof (from Rudin's RCA). And, here are the Theorems 1.27 and 1.34 What I do not understand about Theorem 1.38 is the conclusion that the series $f(x)$ ...
0
votes
1answer
28 views

Jordan outer content of rationals in [0, 1]

How to prove Jordan outer content of rationals in $[0,1]$ is 1 just by definition of Jordan outer content? I mean how to prove this without using "Jordan content of a set is equal to that of its ...
-3
votes
0answers
44 views

If $f$ and $g$ are measurable, is $\{x:f(x)=g(x)\}$ measurable?

My question is if $g$ and $f$ are measurable then $\{f=g\}$ is also measurable. Here is my answer, but I need you to tell me if it is true. Let $\{x\in \mathbb{R}:f(x)˃a\}$ and $\{x\in ...
5
votes
3answers
111 views

Why is the relation “$f=g $ almost everywhere” transitive?

In Rudin's Real and Complex Analysis, it says on pg 27 that If $\mu$ be a measure, define $f\sim g$ iff $\mu(\{x|f(x)≠g(x)\})=0$, where $f,g$ are measurable functions from $X$ to a topological space ...
2
votes
2answers
40 views

Comparing limits of integrals

If $$f_n:X\rightarrow [0,\infty]$$ is a sequence of measurable functions and we know that $$\lim_{n\rightarrow \infty }\int_X f_n \,d\mu=0,\qquad \qquad \tag{$\star$}$$ then can we conclude that ...
-2
votes
1answer
36 views

The Pólya urn model describes a martingale

Suppose an urn contains one blue and one red ball and that we perform the following random experiment: In each round $n\in\mathbb{N}$ we randomly draw a ball If the drawn ball is blue, we replace it ...
1
vote
0answers
31 views

Proof Borel Sigma Algebra

Let $I \equiv \lbrace [- \infty, a[ : a\in \mathbb{R}\rbrace$. Is $\sigma(I)$ Borel's sigma algebra on $\mathbb{R}$? I'm having difficulties proving these statement. I suppose it's not the Borel's ...
-1
votes
1answer
46 views

Are $X$ and $X+Y$ independent, if $X$ and $Y$ are independent? [closed]

As asked in the title? Does the independence of two random variables $X$ and $Y$ imply the independence of $X$ and $X+Y$? If so, what's the easiest way to prove that?
2
votes
1answer
18 views

Marginals of (not necessarily finite) measures

Consider a product of two measurable spaces, $(X,\mathcal{A})$ and $(Y,\mathcal{B})$, and a (not necessarily finite) measure, $\varrho$ on the product space $(X \times Y, \mathcal{A} \otimes ...
3
votes
2answers
45 views

Why is it true that every set in $\mathbb R^n$ can be covered by a countable union of open balls?

Why is it true that every set in $\mathbb R^n(n\geq 2$ can be covered by a countable union of open balls? This is the fact we are using in the definition of measure theory where outer measure of a ...
5
votes
1answer
93 views

Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
0
votes
0answers
45 views

Simple proof check---Exercise from Tao's real analysis notes, Jordan measure

It is a rather simple question, I just want to make sure if my way of doing it works. So the exercise is to proof the monotonicity of Jordon measure (http://en.wikipedia.org/wiki/Jordan_measure). If ...
0
votes
1answer
19 views

Can ratios similar to those related to the surface area of a circle and sphere be applied to determine properties of a 3-sphere?

Applying the strategy of describing the surface area of a circle as a product of the ratio for the surface area of a triangle, reveals a consistency that also applies to the surface area of a cone. ...
0
votes
0answers
21 views

Image of a Borel measure by a local homeomorphism

To simplify matters we assume all topological spaces have countable bases. Let $X$ be a locally compact Hausdorff space, $\mathcal B(X)$ the $\sigma$-algebra generated by the set of open subsets. A ...
1
vote
0answers
8 views

Proving signed measure

I have a small confusion: While proving something to be a signed measure, after proving $\sigma$ additivity, do we need to check that if the series is convergent for some rearrangement of disjoint ...
4
votes
0answers
50 views

Proving completeness of $L^p$

I want to make sure my understanding of the proof is correct. For a Cauchy sequence $\{f_n\}$ in $L^p$, we want to find a $f\in L^p$ such that $f_n\stackrel{L^p}\to f$ Now, skipping the ...
0
votes
0answers
26 views

necessary condition on a non-measurable subset of $[0,1]$ with outer measure 1

Is there any necessary condition that a non-measurable subset of $[0,1]$ with outer measure 1 should satisfy? I was going through the Vitali set with outer measure 1 construction The family F of ...
1
vote
2answers
34 views

Measure Theory: outer regularity

With a couple of friends I am working through some of Tao's Introduction to Measure Theory, which is very good so far. We are stuck on Lemma 1.2.12, Outer Regularity. It says this. Let $E\subset ...
1
vote
0answers
15 views

Subclasses of simple functions dense in $L^2$

Q1. Consider $f\in L^2([0,1], R)$ with $ ||f||^2=\int f(x)^2d\mu(x)$ Consdier a subclass of simple functions $f= \sum_{i=1}^n a_i \chi_{A_i}$ where $A_i\in \Sigma$ (on $[0,1]$) and ...
1
vote
0answers
23 views

Weird behavior of Non-Lebesgue measurable subset of the Smith–Volterra–Cantor set and pointwise convergence of a sequence of simple functions

I came across this (seemingly?) weird behavior of a sequence of simple functions: Let $E$ be the Smith–Volterra–Cantor set and $m: \operatorname{Leb}(\Bbb{R}) \to [0, \infty]$ the Lebesgue-measure. ...
6
votes
1answer
114 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
24 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
523 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
51 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
25 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 ...