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

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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 ...
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18 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
21 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 ...
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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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2answers
43 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
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. ...
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1answer
43 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
43 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
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 $m(f^3) \le m(f^2) m(f)$?
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51 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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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 ...
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1answer
46 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
39 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
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. ...
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1answer
50 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 $\mu(A_n)\to 0$ ...
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1answer
16 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.
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2answers
66 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.
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1answer
35 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 ...
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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, ...
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1answer
18 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 ...
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0answers
29 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 ...
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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 ...
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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 $\mathcal{G}$. ...
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1answer
27 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{ ...
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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: ...
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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 ...
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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 ...
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31 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; ...
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1answer
48 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 ...
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26 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, $K$ ...
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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) = ...
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3answers
30 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
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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 ...
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1answer
45 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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1answer
30 views

Change of variable (or measure)?

Hi Everyone: I am reading a book and there is a kind of "change of variable" they make that I do not understand fully. This is what they do: let $B(x,r)$ be a ball of $\mathbb{R}^{N}$ $(N>1)$, ...
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1answer
48 views

Show that $\frac{1}{x^4 \sin^2 (x) +1} \in L^1([0, \infty))$

This is question 10.20c from Apostol's Mathematical Analysis. Basically, I am trying to show that $$ f(x)=\frac{1}{x^4 \sin^2(x)+1} \in L^1([0,\infty)) $$ I know that for some value $k$, I can ...
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1answer
88 views

“Hidden” axiom of choice?

Let $\mu$ be a measure on $S$ such that: $\mu\left(\emptyset\right)=0$ and $\mu(S)=1$ if $X\subseteq Y$, then $\mu(X)\leq\mu(Y)$ $\mu\left(\{a\}\right)=0$ for all $a\in S$ if $X_n$, ...
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27 views

Lebesgue-Stieltjes measure

Is the following reasonment correct? There is a sort of duality between non-decreasing functions and Borel outer measures. In particular, given a non-decreasing function ...
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1answer
29 views

Set function is a measure

Show that the set function $\mu$, defined on subsets $A \subset \mathbb{N}$ defined by $$ \mu(A) = \sum_{n \in A} 2^{-n}$$ is a measure. The sum of the empty set is defined to be zero and $\mu ...
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2answers
41 views

Measurable Subsets + Caratheodory Measurability

1.) What can go wrong if one assigns a measure to more subsets, especially to all subsets? (I would like to understand the subtleties behind) I imagine the first problem is to give the new subset ...
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1answer
38 views

A well-defined operation on measure algebra

Let $(X,\cal{M},\mu)$ be a measure space, and for $E,F\in \cal{M}$ write $E \sim F$ iff $\mu(E \Delta F)=0$. Let $\widetilde{\cal{M}}$ be the set of equivalence classes in $\cal{M}$ for $\sim$; for ...
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1answer
32 views

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$ is a convex risk measure, but it fails the subadditivity property in order to be called coherent. A mapping ...
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1answer
35 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
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1answer
23 views

continuous integral function

Let $K$ be a countinuous and bounded on $\mathbb{R}^n$ and let $f$ be Lebesque integrable on. a) show that $$g(t) = \int_{\mathbb{R}^n} K(tx)f(x)dx$$ is conituous and well defined. b) suppose that ...
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1answer
22 views

convergence in $L^1$ and convergence $\mu$ a.e imply product convergence in $L^1$

Another old exam problem in measure theory im not sure about. Let $(X,A,\mu)$ be a measure space and $f,g, f_n, g_n$ measurable functions on $X$ such that: $(f_n)$ converges to $f$ in $L^1(\mu)$ and ...
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1answer
16 views

Sum of integrals converge or not!

I have an old exam problems I'm trying to solve $$ \sum_{k = 1}^\infty \int_{-R}^0 \frac{x^k}{k!}dx$$ When $R <\infty$ it seems like dominated konvergence and then change the order of the ...
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0answers
14 views

A basic question on integration by parts in general measure space

Suppose $f_1$ and $f_2$ are measurable functions in a general measure space with measure $\mu$. Is there any standard way to calculate $$\int_{A} f_1 f_2 d\mu$$ where $A$ lies in the sigma algebra
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2answers
52 views

What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
4
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1answer
66 views

Nowhere dense set…

Let $A_n$ be a subset of continuous functions on $[0,1]$ given by: $A_n$ = {$f∈C[0,1]$:there exists $x∈[0,1]$ such that $|f(x)−f(y)|≤n|x−y|$ for all $y∈[0,1]$}. Show $A_n$ is nowhere dense, and use ...
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1answer
22 views

Everywhere continuous extension of a almost everywhere continuous function

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure. If $f$ is continuous outside a set $N$ of $\mu$-measure 0, does there exist an everywhere continuous $g$ such that $f = g$ on $X ...