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

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19 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
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0answers
15 views

A basic question on absolute continuous measures

suppose that $\nu$ and $\mu$ are $\sigma$-finite measures on $(\Omega, F)$ and $\nu \equiv \mu$ (i.e. they are absolutely continuous with respect to each other) consider the set $\{\omega: ...
4
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1answer
76 views

Are “most” continuous functions also differentiable?

Let $A$ be a nonempty open subset of $\mathbb{R}$. Consider a function $f : A \rightarrow \mathbb{R}$. Given that $f$ is continuous, what is the probability that it is differentiable? I suspect it ...
-1
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0answers
16 views

Borel set & measure question [on hold]

Let $ X \subseteq \mathbb{R} $, $\\$ a) Now if $\epsilon \gt 0$ then show that there exist an open set $\mathcal{U} \geq X, $ such that $\lambda (\mathcal{U}) \leq \lambda^*(X)+\epsilon $ . b) Show ...
1
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1answer
60 views

measure space problem.

Let $(\Omega,\mathcal{F},\mu) $ be a probability space. Let $\delta>0$ and for each $n\in \mathbb{N}$. Let $A_n \in \mathcal{F}$ satisfy $\mu(A_n)\ge\delta$. Prove that the set $A_\infty $ ...
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1answer
12 views

Finding the “most” continuous representative of a class of functions equal almost everywhere.

In measure theory, we consider functions to be basically the same if they are equal almost everywhere. It seems crazy, though, to choose any of these as the representative when doing calculations. Why ...
6
votes
2answers
162 views

A sequence of singular measures converging weakly* to a continuous measure

Can anyone provide a sequence of singular (w.r.t. Lebesgue measure) measures $\in\mathcal{M}([0,1])=C[0,1]^*$ converging $weakly^*$ to an absolutely continuous (w.r.t. Lebesgue measure) measure?
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1answer
39 views

$E$ measurable if and only if $E \cap (a,b)$ is measurable for any interval $(a,b)$

We take the definition of measurability to be the following: $E \subseteq \mathbb{R}$ is measurable if for any $\varepsilon > 0$ there is an open set $G$ and a closed set $F$ such that $F \subseteq ...
-1
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1answer
19 views

an example of almost uniformly convergence [on hold]

$x^n\rightarrow 0 $ on [0,1]. does $x^n$ converge almost uniformly to zero. if we take out point 1 or if take out a small set $[1-\epsilon,1]$ why does not it a.u. converge when we take only point ...
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0answers
27 views
+50

Measurability of upper and lower derivatives of Radon measures

Let $\mu$ and $\nu$ be Radon measures in $\mathbb R^N$. Define their upper and lower derivatives by $$ \overline{D}_\nu\mu(x):=\limsup_{r\to0}\frac{\mu(B_r(x))}{\nu(B_r(x))},\qquad ...
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0answers
10 views

scale transformation is invariant for H_1

Consider the subspace $H_1$ of $C_0(0,\infty)$, where $\phi=\int_0^t\dot{\phi}(s)ds$ and $\int_0^{\infty}{\dot{\phi}}^2ds<\infty$. The transformation is $(T\phi)(t)=t\phi(\frac{1}{t})$. How to ...
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0answers
19 views

measurable subset of nonmeasurable set

show that if E is measurable and E⊂P where P is nonmeasurable set in [0,1), then m(E)=0. Can one please tell how to start .. and I have one more question: is the union of m'ble set and non-m'ble set ...
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0answers
13 views

Sufficient condition for equality of two radon measures

Let $ X $ be a locally compact Hausdorff space and let $ \phi_1 $ and $ \phi_2 $ be two Radon measures on X (outer measure means measure and the definition of Radon measure that I am assuming can be ...
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1answer
38 views

Why an integral does not exist?

I am trying to construct a counter example of Fubini Thorem, and for that we need a function $f$ in the product space which is not absolute integrable. So, let ...
5
votes
2answers
127 views

Subsets of $[0,1]$

Suppose we have a closed subset $A\subset[0,1]$ that is not equal to $[0,1]$. Is it possible $mA=1$? Suppose you have an open subset $B\subset[0,1]$ that is dense in $[0,1]$. Is it possible that ...
6
votes
1answer
237 views

How to apply Borel-Cantelli Lemma?

Assume that we are given a sequence of continuous functions $f_n(x)$ on $[0,1]$. How to show the existence of a sequence $a_n$ and a set $A$ with $\mu(A^c)=0$ so that $$ \lim_{ n\to \infty} ...
2
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0answers
40 views

Integral $\int_0^{\pi/3}\log\bigg( \frac{1+2\cos\theta}{2}+\sqrt{\left( \frac{1+2\cos\theta}{2} \right)^2-1}\ \bigg)d\theta.$

Hi I am trying to calculate this integral I given by $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)d\theta. $$ ...
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1answer
3 views

Question about outer regularity and inf

Let $\mu$ be a measure. Suppose for every $\varepsilon > 0$, there exists an open set $U \supset E$ such that $\mu(U) < \mu(E) + \varepsilon$. Then must $\mu(E) = \inf\{\mu(U): U \supset E, U ...
3
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0answers
167 views
+100

Weak and Probability convergences

I have a question about this page, from Topics in Random Matrices Theory, of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
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0answers
8 views

Jordan decomposition of sum of two measures

Let $\mu$ and $\nu$ be finite signed measures. Then by the Jordan Decomposition Theorem, we can write $\mu = \mu^{+} - \mu^{-}$ and $\nu = \nu^{+} - \nu^{-}$ where $\mu^{\pm}, \nu^{\pm}$ are unsigned ...
2
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1answer
35 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
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1answer
58 views

A basic question on limit calculation [duplicate]

How to prove that the following limit exist without calculating its value $$ \lim_{t \to\infty} \int_{0}^{t}\frac{\sin x}{x} dx $$
3
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0answers
49 views

Why is the inverse of the Devil's Staircase not measurable?

I recently did an exercise to show that a monotone function $f:X→ℝ $ is Borel measurable (it even only asked for Lebesgue measurability). On the other hand, the inverse of the Devil's Staircase ...
1
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1answer
22 views

A basic question on measure

Suppose I have a measure in $B(\Bbb R)$ such that for each real number there is a neighbourhood where the measure is zero. Is that measure be necessarily zero measure ? How to prove it ? I can't take ...
1
vote
1answer
41 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
0
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3answers
70 views

a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$ [duplicate]

Hi need some help with this problem: Assume $f : \mathbb{R} → \mathbb{R}$ is a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$, and $f(0) = 0$. Then $f(α) = α ...
1
vote
1answer
31 views

Doubt regarding convergence!

Suppose $f_{n} \to f$ in measure & that there exists a $g \in L^1$ such that $|f_{n}| \le g $ a.e. $\forall $ $n$ . Then, show that: $\lim_{n \to \infty}$ $ \int_{X} |f_{n} - f| \, d\mu$ $ = 0 $. ...
2
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0answers
30 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
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0answers
14 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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votes
2answers
46 views

Inequality with moments

Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here). Is $m(f^3) \le m(f^2) m(f)$?
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0answers
17 views

Decomposition of a measure into series

Let $M(X)$ be the vector space of all complex regular Borel measures on a compact Hausdorff space $X$,$||\mu ||=|\mu|(X)$. Suppose $\mu , \lambda_n \in M(X),n\in N^+ $,$||\lambda_n||=1$.Since ...
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0answers
18 views

$|supp(v)|=0$ implies the existence of $\lim_{\epsilon \to 0}\int_{|\theta -\phi|>\epsilon}\cot{(\pi(\theta-\phi))}dv(\phi)$

Let $v$ be a complex Borel measure on $[0,1]$ and $m$ be the Lebesgue measure. We define the support of measure by $$supp(v) = [0,1]-\cup\{I \subset [0,1]: v(I)=0\}$$ where $I$ is an interval. ...
0
votes
2answers
62 views

A basic question on Riemann sum

Suppose $f$ is a non-negative Riemann integrable function in $[a,b]$. Is this true that $$ \sup_P \sum_{j=1}^{n} |f(c_j)(x_j-x_{j-1})| = \int_{a}^{b} |f(x)|dx$$ where $c_j \in [x_{j-1}, x_j]$. I ...
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1answer
22 views

I want to show one norm is less than or equal to another norm on C([0,1])

Let $|| \ ||_1$ be the norm on $C([0,1])$ defined by $||f||_1 = \int_0^1|f(t)|dt$. a) Show that $||f||_1 \le ||f||_{[0,1]}$ b) Are $|| \ ||_1$ and $|| \ ||_{[0,1]}$ equivalent? For part a) I think ...
2
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2answers
43 views

A basic question on measurability of lim sup and lim inf of a function

Suppose $f: \Bbb R \to \Bbb R$ is a Borel measurable function. I have to prove that $\{x: $f$ \text{ is discontinuous at } $x$\} \in B(\Bbb R)$. So, I am trying to prove that the complement event i.e. ...
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0answers
9 views

What is the relationship of the EMD (Earth movers Distance) and total variation (and other probability measures)?

I was trying to understand different methods for comparing probability distribution and saw the following paper/reference: http://arxiv.org/abs/math/0209021 In it it defines and compares and ...
1
vote
1answer
23 views

Is this set measurable? (Set of points where a sequence converges)

Let $M$ be a manifold. Suppose that $u_n:M \to \mathbb{R}$ are measurable and we have $u_n(s) \to u$ a.e. in $M$. Does it follow that the set $A=\{s \in M : u_n(s) \to u(s)\}$ and $A^c$ are ...
1
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0answers
17 views

Alternatives to Fisher information

The Fisher information matrix is defined as the following: $$\mathcal{I}(\theta)=E[(\frac{\partial \log f(x;\theta)}{\partial \theta})^2]=-E[\frac{\partial^2 \log f(x;\theta)}{\partial \theta ...
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1answer
20 views

If $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$

Suppose $H$ is a Hilbert space. Is it true that if $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$ for any fixed $h\in H$? Certainly if $x_n\to ...
3
votes
2answers
42 views

Boundary of Ball of radius R has zero measure

If $\mu$ is a Radon measure on $\mathbb{R}^n$ and $B_r$ is a closed ball of radius $r$. Why is $\mu(\partial B_r) = 0$? Or how can I prove that there is at least one $r_0 > 0$ such that ...
1
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1answer
26 views

Taking limits of both sides of inequality in Royden and Fitzpatrick

A theorem in Royden and Fitzpatrick's "Real Analysis" relies on taking limits of both sides. First the theorem: Let $E$ be a measurable set of finite outer measure. Then for each $\varepsilon > ...
1
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1answer
41 views

Prove the following sobolev inequality

Let f be a fubction on $[0,1]$ that is continuous and has a continuous derivative f'. show that: $\sup_{0 \leq x,y \leq 1}|f(x)-f(y)|\leq ||f'||_2$. Do not know where to start. Any hint or help is ...
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0answers
37 views

Delta Function as a Conditional Distribution

This is problem 20 from chapter 21 of A Modern Approach to Probability Theory by Fristedt and Gray: Suppose that $X$ is a random variable measurable with respect to a $\sigma$-field $\mathcal{G}$. ...
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1answer
25 views

Unit ball in space of d dimension

If I have a unit ball in space $R^d$ then in how many dimension space its surface will be represented. I know the answer is d-1 but i am unable to convince myself. can anybody give me some intuition. ...
1
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1answer
43 views

Gaining an intuitive understanding of measure & sigma-algebras

Taking my first course in measure theory. Consider an example where $\Omega$={all integers from 1 to 16}={1,...,16} where classes of sets are defined by $C_1$={1, 2, 3, 4, 5, 6, 7, 8} $C_2$={9, 10, ...
1
vote
1answer
111 views

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
3
votes
1answer
48 views

Measure Theory Conjecture

While I was doing some math here, I made this conjecture. Let $f_n:X\rightarrow \mathbb{R}$ be a sequence of measurable functions from the measure space $(X,\mathcal{A},\mu)$ to the measurable space ...
1
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0answers
30 views

Measure inequality implies integral inequality?

Let $f$ and $g$ be non-negative, integrable functions on a measure space with measure $\mu$, and suppose there is some constant $c > 0$ such that for every $t \geq 0$, the inequality $\mu(\{f \geq ...
2
votes
1answer
148 views

a question about caratheodory condition

$f:[a,b]\times\mathbb{R}\rightarrow\mathbb{R}$ is an caratheodory function if $(a)$ the map $z\rightarrow f(t,z)$ is continuous for almost all $t\in[a,b],$ $(b)$ the map $t\rightarrow f(t,z)$ is ...
4
votes
0answers
27 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...