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

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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 $0.99.$

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 ...
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35 views

Suppose $f : [0,1] → R$ satisfies $f(x) − f(y) < x − y$ for all $x,y ∈ [0,1],x > y.$

Suppose $f : [0,1] → R$ satisfies $f(x) − f(y) < x − y$ for all $x,y ∈ [0,1],x > y.$ Show that $f′$ exists almost everywhere on $[0, 1]$ or give a counterexample. Not really sure how to go ...
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1answer
21 views

Probably simple application of Holder or Minkowski for integrals

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 ...
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14 views

Filtration generated by a specific family of random variables

I need help with this problem: Let $X$ be a uniform random variable on $(0,1)$ i.e for every measurable set $A$ $$\mathbb{P}(X \in A)=\int_{A \cap (0,1)}dx$$ Let $X_n= 2^{-n}\lfloor 2^n X \rfloor$ ...
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1answer
38 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 ...
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1answer
52 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 ...
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28 views

Prove that for any measurable set $A ⊆ \mathbb{R}$ $\int_A g_n dm → \int_A f dm.$ [duplicate]

Let $f, g_1, g_2 . . . ∈ L_1(\mathbb{R})$ be non-negative functions. Assume that $g_n → f$ a.e. and $\int_\mathbb{R} g_n dm = \int_{\mathbb{R}} f dm$. Prove that for any measurable set $A ⊆ ...
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1answer
28 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 ...
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0answers
9 views

limit of signed measure that equal $0$

Assume that $\nu$ and $\mu$ are two finite measures on a measurable space $(X,\mathcal M)$. Prove that $\nu <<\mu \iff \lim \limits_{n\to \infty}(\nu-n\mu)^+=0.$ I am seeking help for this.
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2answers
106 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 ...
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2answers
35 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 $m(E) = 1.$ ...
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1answer
33 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| = ...
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0answers
19 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 ...
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1answer
18 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 $J\subset [\alpha,\beta]$. ...
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0answers
11 views

Existence of proper closed set in closed set whose measure different is not zero

Assume that $\mu$ is finite measure on $\Bbb R^n$. Prove that there is a closed set $A \subset \Bbb R^n$ with the property that for each closed $B \subset A$ it follows that $\mu (A\setminus B) \ne ...
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2answers
77 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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23 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 ...
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3answers
66 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 $\mu(E + F) = \mu(\{e + f: e \in E, f \in F\}) = \mu(E)$?
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1answer
53 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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24 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 ...
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17 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 $ ...
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2answers
27 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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1answer
19 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} ...
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1answer
89 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 $A=\{s\in [0,1]:\ u'(s)\ \mbox{exist and}\ u'(s)\neq 0 \}$ ...
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1answer
36 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 ...
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1answer
24 views

Can a regular conditional probability be changed arbitrarily on a null set?

Let $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ and $K$ a probability transition kernel from $\mathbb R$ to $\mathbb R.$ Let $P=K\times\mu$ be a measure on $\mathbb R^2.$ ...
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1answer
21 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 ...
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1answer
18 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!
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1answer
40 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 ...
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1answer
17 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 $B$ ...
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1answer
102 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
23 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 ...
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1answer
21 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 ...
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28 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.
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32 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\} \\ = ...
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1answer
24 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.
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2answers
57 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 $x,y \in \mathbb{R}$. ...
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1answer
12 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 ...
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2answers
47 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 ...
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1answer
22 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
32 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 ...
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21 views

Two integrable borel functions being equal almost everywhere means they have the same integral

Let $(Ω,\mathcal{F},\mathbb{P})$ be a measure space, $X$ and $Y$ two Borel real-valued functions. (a) Show that if $X$ is integrable and $X = Y$ almost everywhere then $Y$ is integrable and $\int X ...
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1answer
60 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 ...
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1answer
27 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
18 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. ...
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1answer
57 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
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1answer
42 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 ...
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1answer
27 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 ...
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1answer
19 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 $f \in L_2$ ...
2
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0answers
34 views

Consider the set $E= \{x∈[0,1]: \sum_{j = 1}^\infty t^j|x−q_j|^{-r} <∞\}$ [closed]

Not sure how to do the following qual problem. Some help would be awesome. Let $q_1,q_2,q_3,...$ be an enumeration of $\mathbb{Q}∩[0,1]$ and let $r,t ∈ (0,1).$ Consider the set $E= \{x∈[0,1]: ...