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

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4
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2answers
162 views

The completeness assumption in Prokhorov's theorem

Originally, I encountered this question on Terence Tao's blog, where the following exercise is presented: Exercise 23 (Implications and equivalences) Let $X_n, X$ be random variables taking values ...
1
vote
3answers
135 views

Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

I am having a hard time with the following real analysis qual problem. Any help would be awesome. Thanks. Suppose that $f \in L^p(\mathbb{R}),1\leq p< + \infty.$ Let $T_r(f)(t)=f(t−r).$ Show ...
1
vote
1answer
194 views

Weak convergence in $C[0,1]$

For a uniformly bounded sequence $(f_n)$ in $C[0,1]$, show that $f_n$ converges weakly to $0$ $\iff $ $\lim \limits_{n \to \infty} f_n(y) =0$ for all $y \in [0,1]$ Is the equivalence true if we do ...
1
vote
0answers
54 views

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$ [duplicate]

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$ Prove that $f = 0$ a.e. Not sure how to start this question. Any ...
1
vote
1answer
49 views

Triangle inequality for integrals with complex valued integrand

This is a step in a lecture note I'm reading. It should be simple because the author considers it obvious but I can't see it. What am I missing? Suppose $U$ and $V$ are integrable over measure space ...
0
votes
1answer
64 views

Question about almost sure convergence.

I am struggling a little to understand almost sure convergence in probability theory. I have taken some general measure theory and there we had abot convergence almost everywhere. Basically it was ...
2
votes
1answer
62 views

the point where all functional are non zero

Let $\{f_n\}$ be sequence of non zero bounded linear functionals on a Banach space X. Show that there is $x\in X$ so that $f_n(x)\ne0$, for all $n\in \Bbb N$. I am confused, non zero functional ...
1
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1answer
34 views

Showing independence of rectangular events…

Suppose I have a sequence of independent random variables $\{X_n, n \in \mathbb N\}$. How do I show formally that $P((X_1,...,X_n)\in A, (X_{n+1},...)\in B) = P((X_1,...,X_n)\in A)P((X_{n+1},...)\in ...
0
votes
2answers
123 views

$\infty + \infty = \infty$?

(The context is a measure-theoretic one.) I know that $\infty - \infty$ is indeterminate, but what about $\infty + \infty = \infty$? It seems this statement is true and if I input it into Wolfram ...
0
votes
2answers
55 views

$m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ Prove that $m(E) = 1.$

Let $E$ be a measurable subset of $[0, 1].$ Assume there is a constant $α > 0$ such that $m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ (Here $m(·)$ denotes Lebesgue measure.) Prove that $m(E) = 1.$ ...
2
votes
1answer
54 views

Prove that $\int|f − g| = \int_{-\infty}^{\infty} μ(F_t △ G_t) dt.$

Let $f$ and $g$ be integrable functions on a measure space $(X,Σ,μ).$ For each $t ∈ \mathbb{R},$ consider the sets $F_t =\{x∈X :f(x)>t\}, G_t =\{x∈X :g(x)>t\}.$ Prove that $\int|f − g| = ...
1
vote
0answers
41 views

Change of variable for Lebesegue Integral

Let $G$ be an absolutely continuous function, $G:[a,b] \rightarrow [c,d]$ and $f \geq 0$ a Lebesegue measurable function in $[c,d]$. I managed to prove that if $f$ is just Borel measurable it holds ...
1
vote
1answer
32 views

Does the pre image of a open interval is a open interval, if the function is absolutely continuous and non decreasing?

Assume that $u:[0,1]\to \mathbb{R}$ is a absolutely continuous (A.C. for short), non decreasing function. Suppose that $u(0)=\alpha$ and $u(1)=\beta$. Take any open interval $J\subset [\alpha,\beta]$. ...
1
vote
2answers
88 views

Are there any cases where $\mathbb E(|X|)<\infty$ and $\mathbb E(X)<\infty$ aren't equivalent?

I often see $\mathbb E(|X|)<\infty$ among the givens in a statement. That made me wonder: why not just demand $\mathbb E(X)<\infty$? In the light of the theorem Let $f$ be measurable. Then ...
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0answers
35 views

Existence of measurable fuction on non-atomic measure space whose integral is infinity

Let $(X,M,\mu)$ be non atomic measure space with $\mu(X)>0.$ Show that there is a measurable function $f:X\to [0,\infty),$ for which $\int f(x)d\mu(x)=\infty.$ No idea at all. I am preparing for ...
1
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3answers
144 views

If $\mu(E) > 0$, $\mu(F) = 0$, is $\mu(E + F) = \mu(E)$?

Let $\mu$ be standard Lebesgue measure. If $E$ is a set of positive measure and $F$ is a set of zero measure, then is it true that $\mu(E + F) = \mu(\{e + f: e \in E, f \in F\}) = \mu(E)$?
4
votes
1answer
72 views

Prove g is Lebesgue intergrable

Let $f$ be Lebesgue integrable on $(0, 1)$. For $0 < x < 1$ define g(x) = $\int_x^1t^{-1}f(t)dt$ Prove that $g$ is Lebesgue integrable on $(0, 1)$. $\int^1_0g(x)dx=\int^1_0f(x)dx.$ I am not ...
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0answers
54 views

A question about uniform convergence in proof that $L^\infty$ is Banach.

I was reading this post Understanding proof of completeness of $L^{\infty}$ and it is mentioned that the sequence $(f_n)_{n\in\mathbb N}$ converges uniformly on a conegligible set $N^C$. Could someone ...
2
votes
0answers
87 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
0
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0answers
44 views

Considering the right Haar measure on the affine group, how does the absolute value come in?

Let $ G $ be the affine group with group action defined by $ (b,a)\cdot(x,s) = (ax+b,as) $ then it is a locally compact group and as such has a Haar measure. In particular the left Haar measure is $ ...
1
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2answers
74 views

Full Lebesgue measure(Dynamical systems)

I am reading a paper and there is a theorem which says: The dynamical system $(D,g)$ is called ergodic on $K\subset D$ if for any saturated subset $A\subset D$, its intersection with $K$ is of either ...
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votes
1answer
24 views

Notation for a collection of sets under a certain condition

I am looking for the notation to describe "A collection of sets that are the union of a finite number of intervals". Is this correct - $A = \{A_i\}_{i \in I}$ where each $A_i = \bigcup_{n \in N} ...
3
votes
1answer
122 views

Properties of absolutely continuous functions

Let $u:[0,1]\to\mathbb{R}$ be a absolutely continuous function. It is know that $u'(x)$ exist almost everywhere and $u'\in L^1(0,1)$. Let $A=\{s\in [0,1]:\ u'(s)\ \mbox{exist and}\ u'(s)\neq 0 \}$ ...
0
votes
1answer
48 views

Random variables and integrals

Could someone please explain how this holds: $\displaystyle \int_{\mathbb{R^n}} f d\mu = \int_{\Omega}f(Y_n)d\mathbb{P}$ Does it use the following proposition? Furthermore how does ...
2
votes
1answer
107 views

Borel set approximation

Suppose $\mu$ is a regular Borel measure on a space $X$. From regularity of $\mu$, every Borel set can be approximated from above by an open set. Can we approximate every Borel set from below by an ...
2
votes
1answer
31 views

Fine Print of Egorov's Theorem

The Egorov's theorem in my textbook requires that the function to be define on a set with finite measure. Why is this necessary, please? Thank you!
0
votes
1answer
64 views

Two questions on Fatou's Lemma

While reading the following paragraph from Real Analysis by Stein (I hope this does not breach any copyright; if so, I have to type it out), two questions occurred to me. In the proof of Fauto's ...
2
votes
1answer
28 views

Measurable set indicator functions - need clarification on a book's statement

A book I'm reading says the following about indicator functions $\chi_A$ : But unless I'm missing something, how can that that be? If $B$ is for example the set $(-2,2)$, $1 \in B$, but since $B$ ...
6
votes
1answer
175 views

Why do we consider measurable function when dealing with abstract integration?

Let $[0,\infty]$ be equipped with the order topology. (That is, it is a subspace of the standard topology on the extended real) Let $(X,\mathfrak{M},\mu)$ be a measure space. Let $f:X\rightarrow ...
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0answers
44 views

Are 'finitely measurable functions' closed under addition?

I'm investigating the properties of Lebesgue measure without countable additivity. That is, I have an outer measure on $\Bbb R$ that satisfies (for subsets $A,B$): $m^*(A)\in[0,\infty]$ $m^*(A\cup ...
0
votes
1answer
108 views

Volume of a cone in an $n$-dimensional ball

Assume that $B$ is an $n$-dimensional ball of radius $R$ centered at the origin, i.e., $B=\{x\in\mathbb{R}^n : \|x\|\leq R\}$. Fix a point $x_0$ in $B$ and $\delta \in (0,\pi)$, and let $C$ be the ...
0
votes
0answers
34 views

Question about measurability in complete measure space

Is it true that For a function $f$ mapping from complete measure space $(\Omega$ ,F,$\mu$) to ($R,L,m$) the lebesgue measurable space in real line. $f$ is measurable iff {$f\le t$}$\in$F.
0
votes
0answers
34 views

Equivalence of two different definitions for $\mu^*$.

For a measure space $(X,S,\mu)$ we have: $\mu^*(A):=\inf\bigg\{\ \sum_{n=1}^\infty \mu(A_n)\ :\ \{A_n\}\subseteq S,\ \{A_n\} \mbox{ disjoint},\ A\subseteq \cup_{n=1}^\infty A_n\bigg\} \\ = ...
0
votes
1answer
82 views

f is measurable iff its coordinate functions are measurable

I am really struggling to connect the sets in $\mathcal{B}(\mathbb{R^n})$ and $\mathcal{B}(\mathbb{R})$. Both inclusions are causing me problems. This questions seems a lot harder than it looks.
2
votes
2answers
79 views

Independence of Random Variables and Distribution Functions

Let $X_1, X_2,\ldots$ be random variables on $(\Omega, \mathcal{A}, \mathbb{P})$. If $\mathbb{P}(X_1 \leq x, X_2 \leq y)=\mathbb{P}(X_1 \leq x)\mathbb{P}(X_2 \leq y)$ for all $x,y \in \mathbb{R}$. ...
1
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1answer
32 views

Elementary question on Dynkin system.

I have got a question while I am solving a problem. Show that $B\setminus A$ is in a Dynkin system if $A$ and $B$ is in the Dynkin system. I was trying to: $B\setminus A=B\cap A^c=B\cap(A^c\cup ...
2
votes
2answers
59 views

Measurability of a function in $\mathcal{B}(\mathcal{C}([0,1],\mathbb{R}))$

The Question i cant answer is, why $\Lambda_a:\mathcal{C}([0,1],\mathbb{R})\rightarrow\mathbb{R}$, given by $\Lambda_a(\omega):=\lambda(\{t \in [0,1]:\omega(t)>a\})$ is ...
2
votes
1answer
126 views

Generalisation of Vitali's covering lemma

In "The geometry of fractal sets", Falconer gives the following generalisation of the Vitali covering lemma as an exercise: Let $\mu$ be any measure on $\mathbb{R}^{n}$ and $E$ a set with ...
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1answer
60 views

$L^p$ norm of a measurable function is bounded by its operation on step functions

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$ $$ \left|\int_0^1 fg d\mu\right|\leq \|g\|_q. $$ Prove that ...
2
votes
1answer
114 views

Suppose that all the functions ${f_n},f$ are integrable. Is $lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Let ${f_n(x)}$ be a sequence of continuous, strictly positive functions on $\mathbb{R}$ which converges uniformly to the function $f(x).$ Suppose that all the functions ${f_n},f$ are integrable. Is ...
1
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1answer
51 views

On the gist of $\sigma(X_1,\ldots, X_n)$

As far as I understand the reason we have $\sigma(X_1,\ldots, X_n)$ all over the probability theory is that it tells us what questions are answerable by $X_1,\ldots, X_n$. Say, we run an experiment ...
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1answer
19 views

Show a set is in the algebra

Assume $A_1, ... ,A_n \in \mathcal{A}$ where $\mathcal{A}$ is an algebra of subsets of a sample space $\Omega$. Set \begin{align} B_i = A_i \setminus (A_1~ \cup~ ...~\cup~ A_{i-1}), i = 1,...,n. ...
1
vote
1answer
290 views

Fat cantor set has positive lebesgue measure

This is probably a duplicate of some other question, but it's not immediately obvious which. The fat cantor set is constructed by removing smaller fractions of the center in each stage of the cantor ...
2
votes
1answer
54 views

a condition given by step functions implies the condition holds for L^q space

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$ Prove $||f||_p\leq 1$. How ...
0
votes
1answer
50 views

Is this subspace dense in $L^{2}(\Omega,\mu)$

Let $(\Omega,\mu)$ be a measure space, and let $X=L^{2}(\Omega,\mu)$ be the complex Hilbert space of square-integrable complex measurable functions on $\Omega$. (Each $f \in L^{2}$ is an equivalence ...
0
votes
1answer
177 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
0
votes
1answer
21 views

do we have $n\mathbb{P}_X([n,+\infty[)\to 0 \quad as\quad n\to +\infty$?

Let X be a random variable. I can't find a rigorous proof to show that $n\mathbb{P}_X([n,+\infty[)\to 0 \quad as\quad n\to +\infty$
2
votes
1answer
28 views

Concentration set of a weak star limit of a sequence of measures.

Let $I=[0,1]$, $\lambda$ the Lebesgue measure and $\mu$ a non negative Borel measure. Assume that $g_n$ is a sequence of non neagative continuous functions satisfying: $$g_n(x)\to 0\ a.e.\ ...
3
votes
1answer
46 views

Is the set of continuous function with Lebesgue zero set a Borel set in continuous space?

Let $D$ be a domain in $\mathbb{R^d}$ and denote the continuous function space on $D$ as $X := C(\overline{D})$ where we can define the $\sigma$-algebra $\mathscr{B}(X)$ of $X$, that is sets in $X$ ...
3
votes
2answers
64 views

Can conditional expectation always be realized in a standard probability space?

Given any integrable random variable $X : (\Omega, \mathcal F, \mathbb P) \to \mathbb R$ (where $(\Omega, \mathcal F, \mathbb P)$ is not necessarily a standard probability space) and a ...