1
vote
2answers
51 views

Sigma-algebra requirement 3, closed under countable unions.

The requirement for sigma-algebra is that. It contains the empty set. If A is in the sigma-algebra, then the complement of A is there. 3. It is closed under countable unions. My question relates ...
0
votes
1answer
24 views

Square of absolute value of a function different than square of function

How come if f is measurable, we might have $|f|^2\neq f^2$? Can you provide an example? I think it is true if f is real.
2
votes
1answer
91 views

Does there exist a nowhere differentiable, everywhere continous, monotone somewhere function?

Is there a nowhere differentiable but continuous everywhere function which is monotone in some small interval however small it is? Until now I have seen only the Weierstrass function and it seems to ...
0
votes
0answers
65 views

A wonderful property of real line!

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this?? I guess $A$ must be an $G_{\delta}$ set which is dense in ...
1
vote
1answer
21 views

Finding an example where a measure is not unique

Let $(X, \mathcal{M})$ be a measurable space. Let $\mu$, $\nu$ be measures defined on $\mathcal{M}$. (a) For $A \in \mathcal{M}$ define $\lambda(A)=\mu(A)+ \nu(A)$. Prove that $\lambda$ is a ...
-1
votes
1answer
47 views

Does $f<\infty$ a.s. imply that $f$ is integrable?

Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $f\colon\Omega\to\overline{\mathbb{R}}$ measurable. Does then $f<\infty$ a.s. imply that $f$ is integrable? I think no, but ...
1
vote
1answer
49 views

Examples for Conditional Expectation (modern probability theory)

I'm in the process of learning about conditional expectation in the framework of modern probability theory. The sudden change brought about by the notion of conditional expectation being a function on ...
1
vote
2answers
39 views

Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.

Show that there exists a not decreasing function $f:(a,b)\rightarrow\mathbb{R}$ continuous on $(a,b)\setminus D$ and discontinuous on $D$ where $D$ is a countably infinite subset of $(a,b)$. This is ...
6
votes
1answer
123 views

Counterexample to “Measurable in each variable separately implies measurable”

Some fellow classmates are preparing for a qualifying exam on real analysis, and asked me for help on the following question: Let $ \ f:[0,1]^2\longrightarrow\mathbb{R}$ be such that: (i) $\ ...
1
vote
0answers
18 views

Is a extension of a premeasure preserves outer-measure generated by the premeasure?

I have proved the follow: Let $X$ be a set. Let $S$ be a semi-ring of subsets of $X$. Let $\mu$ be a premeasure on $S$. Let $\overline{\mu}$ be a premeasure on a ring generated by ...
1
vote
1answer
68 views

Is every Hilbert space an $L^2$ space

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
0
votes
0answers
57 views

Maximum measurable collection and Caratheodory extension

Remember Caratheodory extension theorem: we get an outer measure by extending a premeasure over an algebra. We also have Caratheodory theorem, which give us a measure by restricting that outer measure ...
1
vote
1answer
46 views

Counter example for absolutely continuous measure

I need a example for the following statement: "Given a pair of finite measures $(\mu,\nu)$ on a given measurable space $(\Omega, \mathbb{A})$ is said to have property $P$ if for every $\epsilon >0$ ...
1
vote
2answers
46 views

Example of an integral not converging

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable ...
2
votes
2answers
191 views

Non-measurable set in product $\sigma$-algebra s.t. every section is measurable.

Let $\Omega$ and $\Gamma$ be two nonempty sets and $\mathscr{A}$ and $\mathscr{B}$ be $\sigma$-algebras over $\Omega$ and $\Gamma$, respectively. The product $\sigma$-algebra of $\mathscr{A}$ and ...
1
vote
1answer
17 views

Show that there exitst $f \in L^{1}([0,1])$ such that $\int_{0}^{1}f(x)g(x)dx \nrightarrow 0$

Define $$ g_{n} = n\mathbb{I}_{[0,\frac{1}{n^3}]}(x)\;\; $$ where $\mathbb{I}$ is index function. (if $x \in E, \mathbb{I}_{E}(x) = 1$, otherwise 0) show that there exists $f \in L^{1}([0,1])$ such ...
0
votes
1answer
47 views

Counterexample to: if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ with $\mu(X)=\infty$

We know if $\mu(X)<\infty$, and if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ (can be proved by using Holder's inequality). Is this still true if $\mu(X)=\infty$? Counterexample? ...
3
votes
1answer
101 views

$f$ not measurable, but $\lvert f\rvert$ measurable

Do you know an example of a function $f\colon\mathbb{R}\to\mathbb{R}$ which is not $\mathcal{B}$-measurable but $\lvert f\rvert$ is $\mathcal{B}$-measurable?
3
votes
1answer
75 views

Can we find an example of non-mesuarable set which their outer measure could be computed?

We know there is non-measuarable set and we know every set has outer measure, so can anyone give me an example of a non-measuarable and there outer measure could be computed ?
0
votes
1answer
66 views

Examples of sets which measure cannot be obtained by discretisation

I started reading "An introduction to measure theory" by Terence Tao. On page 23 on a pdf reader (pg 7 in the actual document), we are asked to think of an example of a set $E\subset$ ...
6
votes
2answers
744 views

If $f$ is Lebesgue measurable on $[0,1]$ then there exists a Borel measurable function $g$ such that $f=g$ ae?

If $f:[0,1]\to\mathbb{R}$ is Lebesgue measurable then there exists a Borel measurable function $g:[0,1]\to\mathbb{R}$ such that $f=g$ a.e.?
1
vote
3answers
418 views

Convergence in measure of products

Let $\mu$ be a measure on $(X,\mathcal A)$ and let $f, f_1, f_2,\dots$ and $g, g_1, g_2,\dots $be real valued $\mathcal A$- measureable functions on $X$. Show that if $\mu$ is finite, $(f_n)$ ...
3
votes
1answer
99 views

Outer measure discontinuous from below

I was trying to find an example of an outer Measure which is not continuous from below. These are the definitions I use An outer measure on $X$ is a function $\mu^\ast: \mathcal{P}(X)\to ...
2
votes
1answer
327 views

An example of a generalized Cantor set with positive Lebesgue measure [duplicate]

I want to know if there exist a set $ X\subset \mathbb R$ such that $X$ is $i)$ Perfect $ii)$ Compact $iii)$ Has empty interior $iv)$ Totally disconnected $v)$ Is not countable But $X$ has ...
1
vote
1answer
97 views

Measure, absolutely continuous on boundary

Let $\mu$ be a finite nonnegative Borel measure on $\mathbb R^2_+=[0,+\infty) \times [0,+\infty)$ such that $\mu( \partial \mathbb R^2_+)=0$, i.e. $\mu$ is absolutely continuous on boundary. Is it ...
1
vote
1answer
205 views

If $f_n$ converges uniformly to $f$ on a measure space, show integral of $f_n$ converges to integral of $f$.

Please help me with this problem! Let $(\Omega,\cal F, \mu)$ be a measure space on which $(f_n)$ is a sequence of bounded, measurable, real-valued functions converging uniformly to $f$. If ...
3
votes
1answer
173 views

Counterexample for a non-measurable function?

I am struggling to solve an exercise in my measure theory book and any help for solving it would be appreciated: Let $(\Omega,\mathcal{A},\mu)$ be a measure space and let $f:\Omega \to \mathbb{R}$ ...
0
votes
1answer
81 views

Finite a.e. assumption of Egorov

I am looking for an example to show that the requirement that $f$ be finite a.e. in Egorov's theorem cannot be dropped. I was thinking about $f_n = n$, but here I am not able to see why $f_n$ does ...
10
votes
3answers
229 views

Is there a dense subset of [0,1] of measure 1/2 whose complement is also dense?

I want to find a set $A \subset [0,1]$ so that: $A$ is dense in $[0,1]$ $A^c$ is dense in $[0,1]$ $A$ is Lesbesgue measure $1/2$ (Failing this....I want both sets to be positive measure) My first ...
2
votes
0answers
53 views

Inequality change in $\mathbb{E}[ \max |\cdot|] $ due to $\max$

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $m(W)=1$. Find $m$, a locally-bounded function $f:\mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p \rightarrow ...
1
vote
1answer
80 views

Amenability of abelian and nonabelian groups.

Let $G$ be an abelian group. Is there any probability measure $\mu:\mathcal{P}(G)\to [0,\infty)$ such that for any $A\subseteq G$ and $x \in G$: $$\mu(A)=\mu(xA)$$ How if $G$ is not abelian? (do you ...
0
votes
1answer
23 views

On the existence of functions with a particular convergence

Is the following scenario possible? Provide an example or argue why not. Let $\{f_n\}_{n=1}^{\infty}$ be measurable non-negative functions on $[0,1]$ converging to $f(x)$ pointwise Lebesgue-almsot ...
9
votes
1answer
171 views

Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
1
vote
1answer
79 views

Sequence of continuous fuctions $f_n:[0,1]\rightarrow [0,1]$ s.t. $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ but…

Give an example of a sequence of continuous functions $f_n:[0,1]\rightarrow [0,1]$ such that $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ for every $\varepsilon >0$ but ...
1
vote
1answer
139 views

Hilbert spaces other than $L^2$

From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ ...
5
votes
0answers
233 views

Fourier dimension of sets of positive Lebesgue measure

Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
7
votes
3answers
587 views

Examples of perfect sets.

Let $0\lt a\lt 1$. Can I get examples of of subsets of $[0,1]$ that are perfect sets, contains no intervals and has measure $1-a$. Well, I know by construction the Cantor set is perfect, contains ...
7
votes
1answer
3k views

Lebesgue measurable set that is not a Borel measurable set

exact duplicate of Lebesgue measurable but not Borel measurable BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck... In short: Is there a Lebesgue ...
5
votes
1answer
221 views

Is my counter-example correct?

In my homework for real-analysis I was asked to prove the following statement: On $[0,1]$, for $1\leq{}p<\infty$, If $f_{n}\rightarrow{}f$ a.e. and $||f_{n}||_{p}\leq{}M \space\space\forall\space ...
1
vote
0answers
242 views

Corollary of Lebesgue decomposition theorem and counter-example

Refferring to the Lebesgue decomposition theorem in Lebesgue decomposition theorem and fundamental theorem of calculus there is a corollary when the measure is the Lebesgue measure that states: if ...
13
votes
3answers
475 views

Why does the Continuum Hypothesis make an ideal measure on $\mathbb R$ impossible?

On the page 43 of Real Analysis by H.L. Royden (1st Edition) we read: "(Ideally) we should like $m$ (the measure function) to have the following properties: $m(E)$ is defined for each subset $E$ of ...
1
vote
0answers
66 views

Does the closure of an open set $O$ have the same measure as $O$? [duplicate]

Possible Duplicate: Comparing the Lebesgue measure of an open set and its closure I mean the Lebesgue measure. And one might first look at $\mathbb R^d$ before the an abstract space. ...
31
votes
5answers
1k views

False beliefs about Lebesgue measure on $\mathbb{R}$

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
4
votes
1answer
188 views

An example of a family of atomic measures whose sum is not atomic

I look for a example of family of atomic measures such that their sum is not atomic. A measure $\mu$ on a $\sigma$-algebra $S$ of subsets of $X$ is called atomic if every measurable set of positive ...
6
votes
1answer
473 views

Weird measurable set

In the following, consider the Lebegue measure in $\mathbb{R}^d$. Consider $E\subseteq \mathbb{R}^d$ measurable, with $0\lt m(E)\lt\infty$, such that any measurable subset $F$ of $E$ satisfies ...
1
vote
2answers
300 views

Measure Product Theorem: may non-$\sigma$-finiteness result unique product?

Let $i\in\{1,2\}$. The Measure Product Theorem states that, given the measure spaces $(X_i,\Sigma_i,\mu_i)$, there is at least one product measure $\pi$ such that $\pi(A_1\times ...
13
votes
1answer
226 views

Measurable subset of $\mathbb{R}$ with a specific property

Let $A$ be a subset of $\mathbb{R}$ such that its intersection with every finite segment is Lebesgue measurable. I am looking for an example of such an $A$ with the additional property that the ...
2
votes
2answers
233 views

Dense subset of the plane that intersects every rational line at precisely one point?

It seems there should exist a non-measurable bijection $f: \mathbb{R}\rightarrow \mathbb{R}$. And thus we can obtain a non-measurable graph on $\mathbb{R}^2$ which intersects every horizontal or ...
12
votes
2answers
1k views

Uniqueness of product measure (non $\sigma$-finite case)

Let $(X,\mathscr{A},\mu), (Y,\mathscr{B},\nu)$ be two measure spaces, then we have the product measurable space $(X\times Y, \mathscr{A}\times\mathscr{B})$ where $\mathscr{A}\times\mathscr{B}$ is the ...
16
votes
2answers
750 views

Clarifying the relationship between outer measures, measures and measurable spaces: the converse direction

This is related to my measure theory class, but it's not homework. The motivation behind this post is to understand the exact relationship between the three objects mentioned in the title. ...