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

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0
votes
1answer
106 views

Existence of non Borel-measurable function

Consider two functions f,g defined on the unit interval where f is Borel measurable but g is not. The set where f does not equal g has borel measure 0. I would like to prove that such a function g ...
0
votes
1answer
106 views

Density of linear span of idempotents in $L^{\infty}$

How do I show that the linear span of idempotents is dense in $L^{\infty}(\Omega,\mu)$ where $(\Omega,\mu)$ is a measure space? I don't really have any idea how to do this. Does it involve ...
2
votes
0answers
40 views

show that $\lim_{n\to\infty} \mu^*(\sum_{k=1}^n A_k) = \mu^*(\sum_{k=1}^\infty A_k)$ for outer measure $\mu^*$

Let $\mu$ be a pre-measure on a ring $\mathcal R\subset\mathcal > P(\Omega)$ and $\mu^*$ the outer-measure induced by $\mu$ on $\mathcal > P(\Omega)$. Show that for sets ...
6
votes
2answers
63 views

Writing $f\in L^2([-\pi,\pi])$ as a power series.

Consider the space $L^2([-\pi,\pi])$. I want to show that every function $f\in L^2([-\pi,\pi])$ can be written as a power series. I remember a result that polynomials are dense in $L^2([-\pi,\pi])$. ...
3
votes
0answers
58 views

Basis in $L^2([-\pi,\pi])$

Consider the space $L^2([-\pi,\pi])$. Show that the functions $f_0(x)=1,f_1(x)=x,f_2(x)=x^2,\ldots $ form a basis. The functions are linearly independent (no linear combination adds up to zero). But ...
2
votes
1answer
77 views

Show that set is null set

Let $\mu$ be a measure on $(X,\mathcal A)$. Let $(A_k)_{k\in\mathbb N}$ be a sequence of sets in $\mathcal A$ such that $\sum_{k\in\mathbb N}\mu(A_k)<\infty$. Let $A:=\{x\in X:x\in A_k $for ...
3
votes
1answer
113 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
5
votes
1answer
277 views

Question from “An introduction to measure theory” by Terence Tao [duplicate]

If $(x_α)_{α \in A}$ is a collection of numbers $x_α ∈ [0, +\infty]$ such that $\sum_{α∈A}{x_α} < \infty$, show that $x_α = 0$ for all but at most countably many $α \in A$, even if $A$ itself is ...
4
votes
3answers
170 views

Polynomials are dense in $L^2$

I know that the function $e^{inx}$ can be uniformly approximated on $[-\pi,\pi]$ by polynomials in $x$. I want to use this to show that polynomials are dense in $L^2([-\pi,\pi])$. Suppose that $f\in ...
5
votes
2answers
915 views

$n$th derivative of $(x^2-1)^n$

Define $R_n(x)=\dfrac{d^n}{dx^n}(x^2-1)^n$. Show that $R_n(x)$ is orthogonal to $1,x,\ldots,x^{n-1}$ in $L^2([-1,1])$. Also, what is the value of $R_n(1)$? By definition we have to show that ...
2
votes
1answer
117 views

transition kernel

I've got some trouble with transition kernels. We look at Markov process with statspace $(S,\mathcal{S})$ and initial distribution $\mu^0$. We have a transition kernel $P:S\times ...
2
votes
1answer
57 views

Prove under finite measure equivalent statements

Prove that, if $\mu(\Omega)<\infty$, $(f_n)\subset L^1(\Omega,\mu)$ and $\int_{\Omega}|f_n|d\mu\leq M<\infty$, then ...
1
vote
1answer
32 views

Question about some basic concepts

A $\pi-$system a closed under the formation of finite intersections, $$A,B\in\mathcal{P}\Rightarrow A\cap B\in\mathcal{P}$$ so is this implies that any field is a $\pi-$system, therefore a ...
3
votes
2answers
128 views

Relative sizes of Skorokhod and product topologies on space of sample paths

Let $E$ denote a compact metric space. Let $T$ denote the non-negative reals. Let $E^T$ denote the class of all functions from $T$ to $E$. Let $C$ denote the subset of $E^T$ consisting of càdlàg ...
3
votes
2answers
139 views

Field and Algebra

What is the difference between "algebra" and "field"? In term of definition in Abstract algebra. (In probability theory, sigma-algebra is a synonym of sigma-field, does this imply algebra is the same ...
5
votes
1answer
145 views

Are there some strategies to prove a set has measure zero?

I'm still confused with subsets of $\mathbb{R}^n$ with measure zero. I mean, I know the definition very well: a subset $A$ of $\mathbb{R}^n$ has measure zero if for every $\epsilon > 0$ given ...
0
votes
1answer
260 views

If a function is differentiable almost everywhere, can it be written as an integral?

Consider a function $f:\mathbb{R}^n \to \mathbb{R}$. If $f$ is differentiable with Lebesgue integrable derivative, we may write $$ f(x+y) - f(x) = \sum_{i=1}^p \int_0^1 y_i \nabla f_i(x+ty)dt $$ by ...
0
votes
1answer
127 views

Question regarding Lebesgue outer measure.

Given $m\geq1$, $0\leq s<\infty$, $0<\delta\leq\infty$ and $A\subseteq\mathbb{R}^{m}$ define: $$\mathcal{H}_{\delta}^{s}\left(A\right)=\inf\left\{ {\displaystyle ...
5
votes
1answer
477 views

Differentiation under (measure theoretical) integral sign

I am looking for a citable reference for the result on differentiation under the integral sign for integration against a measure. The result states that if $R \subset \mathbb R$, $(X,\mathcal F, ...
1
vote
0answers
121 views

Subtraction of Probability Measures

I have just read that apparently the following two conditions are equivalent: $$ \int f dP \geq \int f dQ \Longleftrightarrow \int f d(P-Q) \geq 0$$ for $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ and ...
1
vote
1answer
33 views

Map scales and area and volume

What kind of steps I need to prove the following: If we make a map with given scale then the ratio of areas is the square of scale and the ratio of volumes is the cube of scales. I mean, that was ...
0
votes
1answer
333 views

total variation of continuous differentiable function

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function, differentiable on $(0,1)$ and such that $\,f'$ is continuous on $(0,1)$. Prove that $f$ is of bounded variation and $$TV(f,[0,1]) = ...
3
votes
1answer
147 views

Approximating a product-measurable function from below

On page 198 of Dunford and Schwartz, Vol. I, in the proof of part (b) of Lemma III.11.16, the following assertion is made without proof (or reference). Let $(X,\mathscr{X},\mu)$ and ...
2
votes
1answer
378 views

Haar measure of $SO(3)$ obtained from $SU(2)$

I am reading 'Analysis on Lie groups, an introduction' by Faraud and don't understand the following statement … the image by the map Ad of the Haar measure $\mu$ of $SU(2)$ is equal to the ...
3
votes
1answer
319 views

Constructing an increasing function on a set A that is continuous only at the irrational points in A.

The question is the one above and what follows is my attempt. I now need to show that this function is continuous only at the irrational numbers, but I am unsure how to do this.
2
votes
1answer
402 views

tail events and tail sigma-field

I'm working on tail-events. I have a sequence $(X_{n})_{n}$ of random variables an let $\tau$ be its $\sigma$-field. From this I defined $G_n:=\sigma (X_n,X_{n+1},...)$ and $\tau = \bigcap_{n\geq ...
0
votes
1answer
78 views

Baire Category Theorem in a Smooth Manifold

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$. How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using Baire category theorem? and which of the theorem version is ...
1
vote
1answer
77 views

Show $\lim_{n \to \infty} n\cdot m(\{ x \in A | |f(x)| \geq n\}) = 0$

I have to show that $\lim_{n \to \infty} n\cdot m(\{ x \in A | |f(x)| \geq n\}) = 0$, for $(A, \textit{S}, m)$ a measure space and $f: A \rightarrow \mathbb{R}$ an integrable function. Let $A_n = ...
0
votes
1answer
75 views

projection of product space is measurable

Let $(\Omega_i,\mathcal A_i)_{i\in I}$ be a family of measurable spaces. Show that for $\emptyset\neq J\subset I$ the projection $$p^I_J:\prod_{i\in I}\Omega_i\to\prod_{j\in J}\Omega_j, ...
9
votes
2answers
314 views

Does $\mathcal P ( \mathbb R ) \otimes \mathcal P ( \mathbb R ) = \mathcal P ( \mathbb R \times \mathbb R )$?

Obviously the answer's yes if I replace $\mathbb R$ by a countable set. If $\mathbb R$ is replaced by a set $X$ having a greater cardinality, then the diagonal $\{ (x,x) , \ x\in X\}$ is not in the ...
0
votes
2answers
287 views

Symmetric probability measure

Let $\nu$ be a probability measure on $(\mathbb{R},\mathcal{B})$ with $$ \nu((-\infty,t])=\nu([-t,\infty))~\forall~t\in\mathbb{R}. $$ Show that $$ ...
3
votes
0answers
54 views

Show that $\mathcal{N}_{\mu}$ is a $\sigma$-ideal.

Let $(\Omega,\mathcal{A},\mu)$ be a measurable space. Show that $$ \mathcal{N}_{\mu}:=\left\{N\subset\Omega : \exists M\in\mathcal{A}, N\subset M, \mu(M)=0\right\} $$ is a ...
2
votes
0answers
275 views

Prove Lebesgue dominated convergence theorem directly

We can prove that Lebesgue dominated convergence theorem, monotone convergence theorem and Fatou's lemma are equivalent. Almost all the textbooks prove the monotone convergence theorem or Fatou's ...
3
votes
2answers
310 views

fatou's lemma…about the inequality

I'm studying measure theory for the first time, and I just came across Fatou's Lemma. Why isn't it true that for any sequence of functions $\left\{ f_n \right\}$ in $L^+$ we always have that $\int ...
0
votes
2answers
92 views

About assumptions in the monotone convergence theorem

Why is the hypothesis that $\left\{f_n \right\}$ be an increasing sequence essential to the monotone convergence theorem? Could someone provide a nice, easy to understand counterexample if I were to ...
13
votes
1answer
2k views

Monotone convergence theorem by Fatou's lemma

I want to prove the monotone convergence theorem using Fatou's lemma (and its reverse) as exercise, and I need a check; I will use also the following properties of limit inferior and limit superior: ...
1
vote
1answer
684 views

Subset of Measurable Set

Assume we have a measurable set $E$ with positive measure $ \mu (E) > 0$, which means this set is infinite . Let $A\subset E$ and $E-A$ is finite . My Question: Is $A$ measurable ?
6
votes
3answers
599 views

Arithmetic sequence in a Lebesgue measurable set

Let $A\subseteq[a,b]$ be Lebesgue measurable, such that: $m(A)>\frac{2n-1}{2n}(b-a)$. I need to show that $A$ contains an arithmetic sequence with n numbers ($a_1,a_1+d,...,a_1+(n-1)*d$ for some ...
3
votes
2answers
173 views

$\sigma$- ideal

Let $(\Omega,\mathcal{A})$ be a measurable space. $\mathcal{N}\subset\mathcal{P}(\Omega)$ is called a $\sigma$ ideal, if $$ (1)~\emptyset\in\mathcal{N},~~~~~(2) N\in\mathcal{N}, M\subset ...
1
vote
1answer
118 views

Are Compact Sets Separated In a Locally Compact Topological Group

I am studying a proof of the existence of Haar measure on locally compact groups. http://www.albanyconsort.com/HaarMeasure/HaarMeasure.pdf In this proof (at the top of page 7) when proving finite ...
7
votes
0answers
346 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
2
votes
2answers
81 views

Integral construction on $L^1(0,a), a>0$.

I am working on the following problem: Let $a > 0$, $f \in L^1(0,a)$ and define $$ g(x) = \int_x^a f(t) t^{-1} dt, \quad 0 < x \leq a. $$ Show that $g \in L^1(0,a)$ and $\int_0^a ...
2
votes
1answer
46 views

equality for a measure $\mu(F\backslash E)= \mu(F)-\mu(E)$

Studying for Real Analysis I encountered this exercise and I am a bit confused about it. Let $\mu$ be a measure on $(X,M)$, where $M$ is a $\sigma$-algebra on $X$. Show that if $E \subseteq F$ and ...
5
votes
1answer
708 views

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$. So suppose $f\in L^p\cap L^q$. Then both $\int |f|^p d\mu$ and $\int|f|^q d\mu$ exist. For each $x$ in the domain ...
0
votes
1answer
186 views

Well definedness of Lebesgue inner measure

This is for homework: if $A,A'$ are two elementary sets containing $E$, bounded set in $\mathbf{R}^d$, then $m(A)-m^*(A \backslash E)$ is equal to $m(A')-m^*(A \backslash E)$ So far my goal has been ...
2
votes
1answer
35 views

Functions are big only over exponentially small sets

Given $f:[0,1]\rightarrow\mathbb{R}$. Suppose $\mu(\{x\in[0,1]\mid |f_n(x)|>1/n\})\leq 1/2^n$ for all integers $n\geq 1$. Is it true that $\lim_{n\rightarrow\infty}f_n(x)=0$ for almost every $x$? ...
2
votes
0answers
64 views

Integral of continuous function over probability distribution

Let $\mu_f$ denote the probability distribution of $f$ with $f(x)=10x-1$ for $x\in(0,1/2]$ and $f(x)=1$ for $x\in[1/2,1]$. If $g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function, what is ...
0
votes
2answers
59 views

Bernstein set in [0,2]

I have got a question to suggest an example of outer measure that is strictly not additive. I have thought about some special sets such as Bernstein set or Vitali set in an interval to suffice. ...
1
vote
1answer
55 views

The closure of the complement of $A \subseteq \mathbb{R}^d$ with Lebesgue measure zero is $\mathbb{R}^d$?

I have been working on an excercise in measure theory for a few hours now, and although I have learned a lot, the answer to this problem avoids me. It concerns proving the following assertion: ...
5
votes
1answer
78 views

Decreasing sequence of measurable functions

Suppose $f_1(x),f_2(x),\ldots:[0,1]\rightarrow\mathbb{R}$ are measurable functions such that $f_1(x)\geq f_2(x)\geq\ldots$. (infinite sequence) and $\lim_{n\rightarrow\infty}f_n(x)=0$. Is it true that ...