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

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1answer
88 views

Finite subcover of pairwise disjoint open intervals

I have the following exercise: Prove that if $X$ is a countable compact subset of $ \mathbb{R}$, then for any $\varepsilon>0$ there is a finite collection of pairwise disjoint open intervals ...
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0answers
27 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
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1answer
74 views

there is a measurable function $f$ on $X$ such that $|{f(x)}|=1$ for a.a $x \in X$ and $\nu(E)=\int_Efd|{\nu}|$ for any $E \in \mathfrak{M}$

any hints on this problem: Let $\nu$ be a finite signed measure on a measure space $(X, \mathfrak{M})$ and let $|{\nu}|$ be its total variation, prove that there is a measurable function $f$ on $X$ ...
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1answer
585 views

Integral$=-\frac{4}{3}\log^3 2-\frac{\pi^2}{3}\log 2+\frac{5}{2}\zeta(3)$

Hi I have been trying to prove this $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
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1answer
26 views

Showing if functions are equal almost everywhere then

Say $f_n$ and $g_n$ are measurable $f_n=g_n$ almost everywhere for $n=1,2,3,...$ then how can we show that $\sup f_n = \sup g_n$ almost everywhere? I have tried to show that: $$m(\{\sup f_n \neq ...
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3answers
121 views

Hi I was thinking about a problem and have a question: [closed]

Hi I was thinking about a problem and have a question: we know that if $f∈C([0,1])$ for which $∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$! Now my question is: Do we still have the same when we ...
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1answer
46 views

Outer measure induced by a measure

Let $(X, \mathfrak{M}, \mu)$ be a measurable space. Let $\mu^* \ : \ 2^X \ni Y \rightarrow \mu^*(Y)= \inf \{\mu(A) \ | \ A \in \mathfrak{M}, Y \subset A\}$. Prove that $\forall Y \subset X \ \ ...
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0answers
48 views

Proving a theorem in Ergodicity

I read the theorem stated below on Wikipedia (http://en.wikipedia.org/wiki/Ergodicity#Formal_definition). But I do not understand how to prove the equivalence of these different definitions.Any hints ...
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0answers
51 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
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0answers
82 views

Proof that a function is measurable

Suppose $f$ is a joint probability density function of random variables $X$ and $Y$. $Y$ is integrable. I need to prove that the function $g(x) = \int_{\Bbb R} f(x,y)ydy$ is measurable function. I ...
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0answers
29 views

Set of measure zero and $C^{1}$ functions

Does a $C^{1}$ function map a set of measure zero into a set of measure zero?
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1answer
28 views

Certain property of convex functions…

I come to you with yet another qualifying problem we can't seem to solve... Let $f:$ $(0,\infty) \to \Bbb R$ be convex, and let $\lim_{x \to 0}f(x)=0$. Show that $g(x)$ = $f(x) \over x$ is increasing ...
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1answer
151 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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1answer
188 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 ...
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1answer
25 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 ...
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1answer
128 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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1answer
25 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
90 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 ...
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2answers
166 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 ...
14
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1answer
408 views

Integral $\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
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1answer
18 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 ...
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1answer
63 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 ...
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0answers
13 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
61 views

A basic question on limit calculation [duplicate]

How to prove that the following limit exist without calculating its value $$ \lim_{t \to\infty} \int_{0}^{t}\frac{\sin x}{x} dx $$
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0answers
98 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
26 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 ...
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3answers
89 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 ...
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1answer
35 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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1answer
54 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 ...
2
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0answers
49 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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1answer
36 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 ...
1
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1answer
89 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 ...
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0answers
25 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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1answer
36 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 ...
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0answers
33 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 ...
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0answers
23 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. ...
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1answer
34 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 ...
2
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0answers
49 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
27 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 ...
2
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2answers
76 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. ...
4
votes
2answers
85 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 ...
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1answer
43 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. ...
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1answer
60 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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1answer
67 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 ...
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2answers
68 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 $m(f^3) \le m(f^2) m(f)$?
4
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1answer
61 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 ...
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2answers
86 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 ...
1
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1answer
67 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, ...
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2answers
54 views

Is $C(X)$ dense in $L^p$?

Let $X$ be locally compact Hausdorff. Let $\mu$ be a complete measure on $X$. Is $C(X)$ dense in $L^p(\mu)$?
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1answer
43 views

Where to find Geman 1995's proof on Changes of Numaraire?

Geman, H., El Karoui, N., Rochet, J.C. (1995) published paper "Changes of Numeraire, Changes of Probability Measures and Pricing of Options", on "Journal of Applied Probability " vol 32, pg 443-458. ...