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

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11
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1answer
253 views
+400

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

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 ...
0
votes
1answer
7 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 ...
0
votes
0answers
11 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 ...
1
vote
0answers
11 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 ...
0
votes
0answers
61 views

A basic question on limit calculation [duplicate]

How to prove that the following limit exist without calculating its value $$ \lim_{t \to\infty} \int_{0}^{t}\frac{\sin x}{x} dx $$
3
votes
0answers
58 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
vote
1answer
23 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 ...
0
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3answers
79 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
votes
1answer
40 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
votes
0answers
35 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} ...
1
vote
1answer
28 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
vote
1answer
45 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 ...
1
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0answers
18 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 ...
0
votes
1answer
24 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 ...
1
vote
0answers
11 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
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0answers
19 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. ...
1
vote
1answer
24 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
votes
0answers
23 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 ...
0
votes
1answer
22 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
votes
2answers
45 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
45 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
vote
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
vote
1answer
44 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 ...
1
vote
1answer
44 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 ...
0
votes
2answers
47 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 $m(f^3) \le m(f^2) m(f)$?
3
votes
1answer
53 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 ...
0
votes
2answers
63 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
vote
1answer
49 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, ...
0
votes
2answers
42 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 $L^p(\mu)$?
1
vote
1answer
14 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. ...
1
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1answer
56 views

If $\mu(A_n)\to 0$ then $\int_{A_n} f d\mu \to 0$.

Let $(X, M, \mu)$ be a measurable space. I'm trying to prove the following statement: If $f \in L^p$, $1<p<\infty$ and $\{A_n\}$ is a sequence of measurable sets sucht that $\mu(A_n)\to 0$ ...
1
vote
1answer
18 views

$\|f\|_4 \le C \|f\|_2$ for all $f \in L^4([0,1])$

Does there exist a constant $C$ such that $\|f\|_4 \le C \|f\|_2$ forall $f \in L^4([0,1])$? I haven't been able to find this $C$, so I'm not sure if it exists or not.
0
votes
2answers
68 views

$L^p(\mathbb R) \subseteq L^q(\mathbb R)$?

Is it true that $L^p(\mathbb R) \subseteq L^q(\mathbb R)$ for $1 \le p <q <\infty$? I haven't been able to find a counterexample, so I'm startig to suspect it is true.
3
votes
1answer
38 views

Definition of completion of a measure space

On a measure space $(\Omega,\mathcal{A},\mu)$ a completion of a measure is defined as: $\{A: A_1\subset A\subset A_2$ with $A_1,A_2\in\mathcal{A}$ and $\mu(A_2\backslash A_1)=0\}$ I'm trying to show ...
0
votes
1answer
36 views

$\int f d\mu<\infty$ iff $\sum_{n=0}^\infty 2^{-n} \mu(\{x \in X : f(x) \geq 2^{-n}\})< \infty$.

I have to prove this, but I really don't have any idea of how to start, I don't know which result or technique I could use. I would appreciate any hint or idea to prove this. Thank you. Let $(X, ...
0
votes
1answer
19 views

If $f:X\to\mathbb{C}$ is integrable, is it $|\int f|\le \int |f|$?

Let $(X,S,\mu)$ be a measurable space. We say that $f:X\to\mathbb{C}$ is integrable if $Im(f)$ and $Re(f)$ are integrable, and we put $\int f :=\int Im(f)+i\int Re(f)$. Is it true that $|\int f|\le ...
4
votes
0answers
37 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 ...
2
votes
1answer
15 views

finding conditions for certain limits of integrals…

let $g$ continuous on $[0,1]$. find conditions on function $g$ that are equivalent to $lim_{n \to \infty} ||g^nf||_{2}=0$ for all $f$ in $L^2(0,1)$ we are completely stuck on this one. Tried some ...
1
vote
0answers
39 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 $\mathcal{G}$. ...
1
vote
1answer
28 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
2
votes
2answers
45 views

$\lim_{k\rightarrow \infty}\frac{2^k}{\gamma}\log\mathbb{E}[e^{-\gamma \frac{X}{2^k}}]$

I am trying to find a limit for this expression $$\lim_{k\rightarrow \infty}\frac{2^k}{\gamma}\log\mathbb{E}[e^{-\gamma \frac{X}{2^k}}]$$ I have so far found these bounds: ...
3
votes
1answer
21 views

Convergence in average on every set implies convergence?

Let's say we're working in a measure space $(X, \mathcal{B}, \mu)$, and let $f_n, f$ be measurable. Suppose I have that, for any measurable set $E$, $$ \int_E f_n d \mu \to \int_E f d \mu $$ Does that ...
2
votes
1answer
23 views

Usefulness of criterion for weak convergence

I am currently reading the book Convergence of Probability Measures by Patrick Billingsley, and I came across the following theorem: Theorem. Let $(S,\rho)$ be a metric space, and $B(S)$ be the Borel ...
0
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0answers
34 views

$\lim\limits_{n\to\infty}\displaystyle\int_X n\log((1+(f/n)^{\alpha})d\mu$

suppose $\mu$ is a positive measure on $X$ and $f:X\to[0,\infty]$ is measurable with $\int_Xfd\mu=c$, where $0<c<\infty$ and let $\alpha$ be a constant, prove that; ...
3
votes
1answer
50 views

Convergence in measure of integrable functions implies limit is integrable?

I'm going through my handwritten notes for my upcoming exam (so not homework) and the above was stated but not proven in class. The full statement is a little different, but the above part is the only ...
0
votes
0answers
29 views

Probability theory in (classical) cryptography

In (classical) cryptography we have the formal definition of a cryptosystem that is a quintuple $(M,C,K,e,d)$ where $M$ is the (finite) set of plaintexts, $C$ is the (finite) set of ciphertexts, $K$ ...
1
vote
0answers
25 views

Finding the norm of a linear operation.

I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ...
0
votes
3answers
33 views

how to use holders inequality to show lq is a subspace of lp

suppose the measure of X is finite, I want to show Lq(X) is a subspace of Lp(X), where 1<=p<=q<=infinity I know I need to use holders inequality, but I am not sure how do I use it Thanks
0
votes
1answer
21 views

interacting probabilitys

Find two absolutely continuous probability measures $\mu(x)dx$ and$\nu(x)dx$ with finite second moments. Such that the function $f(t)$ we have that $\dfrac{d^{2}}{dt^{2}}f(t)<0$ where ...