# Tagged Questions

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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### 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 $\epsilon >0$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### $f$ 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?
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### 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 ?
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### 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 $E\subset$ ...
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### 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 ...
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### 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.
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### 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.?
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### 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, $(f_n)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $G$ ...
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### 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.
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 ...
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 ...