1
vote
1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
2
votes
1answer
24 views

Absolutely continuous probability measures example

I was given the following definition: Then this example: It is said that $\mathbb P_1\ll\mathbb P_2$ , but I don't really see it.Please help.
2
votes
1answer
66 views

Does every nonmeasurable set split into a measurable subset and a purely nonmeasurable subset?

Being curious I'm wondering: Suppose you're given a continuous function over a Borel space. Then the preimage of every open is measurable. However, while the preimage of every neighborhood of some ...
1
vote
1answer
54 views

Definition of the total variation of a measure: countable partitions versus finite partitions

The total variation according to Rudin is defined as: $$|\mu|(E):=\sup_{\bigcup_{k\in\mathbb{N}}E_k=E}\sum_k|\mu(E_k)|$$ where the supremum is taken over all countable partitions. Now I'm reading in ...
11
votes
2answers
188 views

Do differentiable functions preserve measure zero sets? Measurable sets?

Consider Lebesgue measure on $\mathbb{R}$ and let $f:\mathbb{R}\to\mathbb{R}$ be differentiable. Does $f$ necessarily preserve measure zero sets? Does $f$ necessarily preserve measurable sets? ...
1
vote
1answer
21 views

Necessity of a hypothesis in Scheffé's lemma

Scheffé's lemma states that if $f_n$ is a sequence of Lebesgue integrable functions (i.e. $f_n \in L_1$) that converges almost everywhere to another integrable function $f$, then $\int |f_n - f| \, ...
1
vote
0answers
23 views

Counterexample for converse about measurable sections

On page 67 of Jacod and Protter, Probability Essentials, it is stated that: Theorem 10.2 Let $f$ be measurable: $(E \times F, {\cal E} \otimes {\cal F}) \to (\mathbf R, {\cal R})$. For each $x \in ...
1
vote
1answer
50 views

Counterexample to Marcinkiewicz

I have a version of Marcinkiewicz: Let $(X,\mu)$ and $(Y,\nu)$ be measure spaces and let $1<p_1 \leq \infty$. Suppose that $T$ is a mapping from $L^1(X,\mu) + L^{p_1}(X,\mu)$ to $\mu$- measurable ...
1
vote
1answer
39 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
1
vote
1answer
56 views

weak star convergence of signed measures vs convergence in Fortet-Mourier norm

There is a norm for signed measures given by $$||\mu||_{FM}=\sup_{f\in \mathrm{Lip}_1(X),|f(x)|\leq 1}\langle f,\mu\rangle.$$ This is usually called Fortet-Mourier norm (or more often metric, but it ...
1
vote
2answers
81 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
27 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.
3
votes
1answer
129 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
1answer
110 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ 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
32 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
78 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
42 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 ...
7
votes
1answer
153 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
22 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
72 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
64 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
70 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
47 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 ...
3
votes
2answers
310 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
58 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
115 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
79 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
69 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 $E\subset$ ...
1
vote
1answer
649 views

Example of Converge in measure, but not converge point-wise a.e.?

Can anyone give an exam of Converge in measure, but not converge point-wise a.e.? And also for the converse part, professor asks us to prove "pointwise a.e. implies converge in measure", but think ...
7
votes
3answers
729 views

Examples of uncountable sets with zero Lebesgue measure

I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set. Thanks.
7
votes
2answers
888 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
555 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
110 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
371 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 ...
2
votes
1answer
114 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
236 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
197 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
85 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
270 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
54 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
92 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
172 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
141 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 $G$ ...
3
votes
4answers
686 views

A set that it is uncountable, has measure zero, and is not compact

I want a example of a set that it is uncountable and has measure zero and not compact? Cantor set has these properties except not compactness.
5
votes
0answers
237 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
649 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 ...