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

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1answer
19 views

Dense subset of $[0,1]$ with Lebesgue measure $\epsilon$

We wish to find a Lebesgue measurable subset of $[0,1]$ that is in dense in $[0,1]$ with measure exactly $\epsilon$, where $\epsilon \in (0,1)$. My idea is to let $I=(0,\epsilon)$ and let ...
0
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0answers
35 views

classifying topological spaces by measures

While looking at some spaces, I happend to know,that in some spaces(like $\mathbb R^n$) Null sets have topological properties(defining the Algebra by the open sets)! some examples: in $\mathbb R^n$ a ...
1
vote
1answer
22 views

Banach spaces not isomorphic to $\ell^p(S)$?

We know that every Hilbert space is unitarily equivalent to $\ell^2(S)$, for a set $S$ of suitable cardinality. Is there a Banach space which is NOT isomorphic to $L^p(X)$, for any $1\leq p \leq ...
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0answers
10 views

Wiener Measure of $E=\{B_{t}: B_0=x$ and $B_{t}\in \partial B_{0,r}\}$ in integral form

The Wiener measure of the event $E=\{B_{t}: B_0=x\in (B_{0,r})^{c} $ and $B_{t}\in \partial B_{0,r}$ for some t $ \}$,is the measure of all the possible Brownian paths that start from x and hit $ ...
1
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1answer
38 views

A Question on Sets of Full Outer Measure

I came across this problem whilst studying for a comprehensive exam in real analysis; for reference, see Exercise 1.24(A) in Folland's Real Analysis; it's a modification of that. Consider the unit ...
1
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0answers
27 views

When is an outer Borel regular measure (inner and outer) regular?

Let $X$ be a topological space and $\mu$ an "outer" Borel regular measure on $X$ (for all $A\subset X$, there is $B$ Borel with $\mu(A)=\mu(B)$). Assume that $X=\cup _{i=1}^\infty U_i$, where each ...
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0answers
24 views

Prove something is a signed measure

Given a measure space $(X,\mathcal{M},\mu)$ and a measurable function $f:X\rightarrow \overline{\mathbb{R}}$ such that at least one of $f^+$ or $f^-$ is integrable, show that ...
1
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0answers
38 views

$f_x(y) = f(y-x)$, $L^p(\mathbb{R}^n)$ [on hold]

Let $x \in \mathbb{R}^n$ and $f \in L^p(\mathbb{R}^n)$, $f_x$ function on $\mathbb{R}^n$ defines $f_x(y) = f(y-x)$.Let fix $f$ and $1 \leq p < \infty$. Prove that is mapping $x \mapsto f_x$ ...
1
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0answers
22 views

Proof that Lipshitz function has a primitive

I was doing exercise 5 of this exercise sheet: http://didel.script.univ-paris-diderot.fr/claroline/backends/download.php?url=L1RENi5wZGY%3D&cidReset=true&cidReq=31UKMT42 And I don't know how ...
1
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0answers
9 views

Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
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0answers
12 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
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0answers
9 views

Two Radon measures and mutual singularity

Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$. Show that $\mu$ and $\lambda$ are mutually singular iff $D(\mu,\lambda,x)=\infty$ for $\mu$ almost all $x \in \mathbb{R^n}$. I have looked ...
2
votes
2answers
33 views

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family ...
1
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0answers
78 views

Riemann implies Lebesgue integrablility on $\mathbb{R}^n$, prove $f(x)$ continuous at x where $g(x)=G(x)$

Let $f:[a_1,b_1]\times \cdots \times[a_n,b_n] \rightarrow \mathbb{R}$ be Riemann integrable. Prove that is $f$ Lebesgue integrable. Proof: $$Q:= [a_1,b_1]\times \cdots \times [a_n,b_n].$$ For simple ...
-1
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0answers
26 views

Darboux integrable, $f$ continuous at x where g(x)=G(x) [duplicate]

$f:[a_1,b_1]x[a_2,b_2]\rightarrow \mathbb{R}$ that is Riemann integrable, and let $g(x),G(x)$ functions with property $g(x)\leq f(x) \leq G(x)$, g=G a.e.! G(x), g(x) are obtain from proof Riemann int ...
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0answers
34 views

Integrable function $f$ and simple function $\phi$ such that $ \int{|f-\phi|} \> d\mu < \epsilon.$

I am seeking solution verification for the following problem. Suppose $f$ is an integrable function. We wish to show that there exists a simple function $\phi$ such that $$ \int{|f-\phi|} \> d\mu ...
0
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1answer
32 views

Convergence in measure theory

Let $u_n$ be a sequence converging to $u$ in $L^2$. Let $f(t)$ be a bounded continuous function. Can I say that $f(u_n)$ converges to $f(u)$ in $L^p$ for every $1<p<\infty$?
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0answers
30 views

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$. [duplicate]

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$. From here http://math.stackexchange.com/a/878635/140308 (proof attempt is there too) Sorry ...
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0answers
34 views

Subset Lp space. [on hold]

Let S closed vector space subset $L^1$($\mu$), where $\mu(X) < \infty$. Assume $f \in S \Rightarrow f \in L^p(\mu)$, for some $p>1$. Prove that $\exists p>1$ so that $S \subset L^p(\mu)$?
0
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1answer
23 views

A field being a sigma field if and only if it's a monotone class

The exercise is as follows: "The limit of an increasing (or decreasing) sequence An of sets is defined as its union ∪nAn (or the intersection ∩nAn). A monotone class is defined as a class ...
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0answers
10 views

Is every frame homomorphism induced by a measurable function?

Let $M$ be the Lebesgue measure algebra of the unit interval $[0,1]$, i.e. equivalence classes of Lebesgue measurable sets modulo sets of measure $0$. This is a complete Boolean algebra, hence in ...
0
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1answer
35 views

A tricky integral with vanishing domain

I would love to have the following result, however I got no clue if it is even true! Let $B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\}$ for some sequences ...
1
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2answers
31 views

Set of points at which sequence of measurable functions converge (another approach)

Question is to prove that : Set of all points at which a sequence of measurable functions converge is a measurable set.. What i have tried is as follows : We are looking at the following set : ...
0
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1answer
21 views

Vitali set of outer measure 1

How to construct a Vitali set of outer measure 1. I couldn't understand the argument given here. Isn't there any easier way? I would also like if someone explains that to me. Thank you in advance!
0
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0answers
17 views

Finiteness of the lower integral implies finiteness a.e. of the function

I want to prove that if a function $f$ is $\mu$-measurable, $f\geq 0 $ $\mu$-a.e., then the integral of $f$ exists, that is its upper and lower integrals coincide. I've found the proof in Modern and ...
0
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1answer
41 views

If $f$ is $+\infty$ on a set of positive measure and the integral exists in $[-\infty,+\infty]$, must the integral be $+\infty$?

Suppose $(X,\mathcal{M},\mu)$ is a measure space and $f$ a measurable function from $X$ to $[-\infty,+\infty]$. Suppose that $$\int_{X}f\ d\mu$$ exists in $[-\infty,+\infty]$, and that $X$ contains a ...
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1answer
34 views

Counterexamples in measure theory

Can you suggest me a book which primarily deals with counter-examples in measure theory? Thank You in advance!
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0answers
20 views

Borel measure which is not regular

I need an example where a Borel measure is not regular. I already proved that any finite Borel measure is regular and tight. The examples of irregular measures given here are too rigid and not so ...
3
votes
0answers
59 views

Why is the value assigned to a gauge integral well defined (unique)?

Why is the value assigned to a gauge integral well defined (unique)? If we would have given a net (so an underlying order that happens to be directed), then the limit would be unique given a ...
1
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2answers
24 views

Totally disconnected measurable set with positive measure

Could we find a totally disconnected set of the real numbers which is Lebesgue measurable and has positive measure?
1
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1answer
24 views

$\sigma(X_1,\dots, X_n) = \sigma(S_1,\dots, S_n)$ generated by partial sums

is true that $\sigma(X_1,\dots, X_n) = \sigma(S_1,\dots,S_n)$ where $S_n=\sum_{i=1}^n X_i$ in general or I have to impose additional restrictions to the random variables (for instance, independence)? ...
4
votes
2answers
109 views

The set $E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\}$ does not contain all irrational numbers in $[0,1]$

Let $q_1,q_2,q_3,...$ be an enumeration of $\mathbb{Q}\cap[0,1]$ and let $r,t \in (0,1).$ Consider the set $$E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\} $$ (a) Show that $E\neq ...
2
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0answers
31 views

Asymmetry in definition of regular measure

In a Borel measure space $(X, \mathcal{B}, \mu)$, $\mu$ is outer regular at $E$ if \begin{equation} \mu(E) = \inf_{U \textrm{ open}} \{\mu(U): U \supseteq E\} \end{equation} and ...
3
votes
2answers
77 views

Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
0
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2answers
56 views

Uniform integrability of a function in $L^1$

A collection of functions $(\phi_i)_{i\in I}\in L^1(\mu)$ is called uniformly integrable if given $\epsilon>0$ there exists $\delta>0$ such that : $$\int_E|\phi_i|d\mu<\epsilon~~~~\forall ...
-4
votes
1answer
21 views

The equivalent condition of the almost surely convergence [on hold]

$X_n\rightarrow X$ a.s. if and only if, given $\epsilon>0$ and $\delta>0$, there exists $n(\epsilon,\delta)$ such that $\mathbb{P}\{|X_n-X|\geqslant\epsilon \mbox{ for some } n\geqslant ...
2
votes
1answer
43 views

Why $m(B^n(0, r)) = c_nr^n$?

(Bear with me: I realize this is quite basic question, but I'm a little loss at how to search for an answer). Anyway: I came across a real analysis proof which uses a property that (as far as I can ...
0
votes
1answer
37 views

Measurable function that's defined almost everywhere

If $(X, \Sigma, \mu)$ is a complete measure space, and $f$ is a function that is defined almost everywhere, can I use the language that $f$ is measurable? What does it mean for this function that is ...
2
votes
1answer
24 views

Can we integrate a measurable function defined on a conull subset of a complete measure space?

Suppose $(X, \Sigma, \mu)$ is a complete measure space, and suppose $f$ is a measurable function with domain of $f$ the set $X \setminus N$ for a measurable set $N$ of measure $0$. Does it make sense ...
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0answers
53 views

Show that $\int_{\mathbb{R}^n}f_1f_2 …f_n dx_1 …dx_n ≤ (I_1 …I_n)^{1/(n−1)}.$

For $\quad k = 1,2,...n,\quad$ let $\quad\mathbb{R}^k = \mathbb{R},\quad f_k(x_1,...,x_{k−1},x_{k+1},\ldots,x_n)\quad$ be a nonnegative measurable function on $\quad\mathbb{R}_1\times\ldots\times ...
3
votes
0answers
61 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
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0answers
34 views

The restriction of an open bounded linear operator

I need some help with this question. Let $X$ be a Banach space and $T:X \to X$ be a bounded linear operator. Suppose that $T$ is open, and $X_0$ be a closed subspace of $X$. The restriction $T_0$ of ...
0
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0answers
24 views

Distribution of an upper limit of a sum of random variables

Let $\{X_{n,j},n\geqslant1,j\geqslant1\}$ be independent and identically distributed random variables. Denote $S_{n,k}=\sum_{j=1}^kX_{n,j}$. Let $\{Z_n,n\geqslant1\}$ be a sequence of random variables ...
2
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3answers
45 views

Example of sequence of measures?

Can you give me some examples of sequence of measure that converge to a measure? (I am reading the topic of weakly convergence of measure, and convergence of random variables in distribution)
1
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1answer
31 views

Existence of ergodic joining

Let $\underline{X}=(X,\mathcal{B},\mu,T)$ and $\underline{Y}=(Y,\mathcal{B},\mu,S)$ be ergodic measure preserving systems on Borel probability spaces. A joining of $\underline{X}$ and $\underline{Y}$ ...
1
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1answer
56 views

Sequence of measurable functions converging a.e. to a measurable function?

I understand if $(X, \Sigma, \mu)$ is a measure space, and we have a sequence of measurable functions $f_{n}$ such that $\lim \limits_{n \to \infty} f_{n}$ exists almost everywhere d$\mu$ (a.e. ...
3
votes
2answers
42 views

Convergence in $L^p$ by using Holder's inequality

Let $1\lt p \lt \infty$ and $f\in L_p[0,\infty )$. Show that a) $$\left\vert\int_0^x f(t)\,dt\right\vert\le\|f\|_px^{1-\frac{1}{p}},$$ for $x\gt 0$. b) $$\lim_{x\to \infty} ...
3
votes
1answer
92 views
+50

Showing a certain sequence is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$

The problem is to show $$\left\{\frac{1}{\pi^{1/2}(i+z)}\left(\frac{i-z}{i+z}\right)^n\right\}_{n=1}^{\infty}$$ is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$ In another exercise, I have ...
2
votes
1answer
25 views

Confusion on statement of Fubini's theorem for characteristic function of measurable set

I'm having trouble understanding what this theorem is saying. Theorem. Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ be a complete measure space and suppose $E \in \overline{\Sigma ...
2
votes
3answers
29 views

Finite additivity in outer measure

Let $\{E_i\}_{i=1}^n$ be finitely many disjoint sets of real numbers (not necessarily Lebesgue measurable) and $E$ be the union of all these sets. Is it always true that $$ m^\star (E)=\sum_{i=1}^N ...