# Tagged Questions

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104 views

### Volume of normal cone of a simplex (at a vertex)

This question is related to an orthant-type simplex in $\mathbb{R}^n$, which can be defined as
$$ S = \{x\in\mathbb{R}_{+}^n: \sum_{i=1}^nx_i \leq 1\}=\overline{\mbox{conv}}(0,e_1,\ldots,e_n). $$
For ...

**1**

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**0**answers

97 views

### Make this marginalization statement rigorous

Intuition tells me that
$$ p(x\,|\,y) = \int p(x,\theta\,|\,y) \; d\theta$$
by the "law of marginalization", pretty much for any object $\theta$.
I would like to make this statement rigorous, ...

**3**

votes

**1**answer

153 views

### Extension of a bilinear functional

Does any one know an example of a bilinear functional $B:C(X)\times C(Y)\to {\bf R}$ ($X$ and $Y$ are open subsets of Euclid spaces) which cannot be extended continuously to a measure $\mu:C(X\times ...

**3**

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**1**answer

221 views

### Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space
$(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...

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**2**answers

186 views

### Relationship of Euler product to coprimality densities for arbitrary sets of primes

Continuing the curiosity of my last couple questions: Is it the case that for every set of primes $F$, the asymptotic density of the integers coprime to all of $F$ is $\displaystyle \prod_{p \in F} (1 ...

**1**

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**1**answer

51 views

### Additivity of asymptotic density of periodic sets

It is well known that the asymptotic density of an infinite union of disjoint sets of integers may not be the sum of their individual asymptotic densities.
Can this failure of countable additivity ...

**3**

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**1**answer

123 views

### For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1

This is related to a question asked on mathstackexchange http://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo. This question is ...

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221 views

### Question on separability of a measure

Following this question here this question come to mind.
Consider a measured σ-algebra $(S,\mu)$ . Assume that μ is normalized to have total weight 1, and that S is complete (contains all subsets of ...

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**1**answer

48 views

### Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...

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51 views

### Approximate of binary function by indicator functions of one variable

Let $(X,F,u)$ and $(Y,G,v)$ be two probability spaces.
I know a result that the indicator functions of product measurable sets can be approximated by indicator functions of one variable. That is, for ...

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**0**answers

72 views

### Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
...

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**2**answers

107 views

### How does one prove that $L_1(\mu)$ is weakly sequentially complete for any measure?

It is a theorem of Steinhaus that for any finite measure $\mu$, the Banach space $L_1(\mu)$ is weakly sequentially complete. Using the Radon-Nikodym theorem one can extend this easily to ...

**1**

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**1**answer

115 views

### Laplacian on space of measures

Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm.
The Laplace-Belrami-Operator $\Delta$ on $X$ with ...

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**1**answer

185 views

### Minimal generator of an algebra or a sigma-algebra

I may be asking a trivial question, but I am a bit confused about it. I have tried to search for the concept of a minimal generator of an algebra or a sigma-algebra on a set, but have found this ...

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**0**answers

33 views

### Where to read about this kind of “measure of irredundancy” of a set from a family of sets?

Studying a very practical problem from psychometrics, I encountered the following construction.
Let $(X,\mu)$ be a measure space; if preferred, you can presume $\mu$ is a probability measure. In any ...

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**1**answer

172 views

### A very natural question in weak* topology [closed]

Can you provide me a counter example for this.
Suppose that I have a sequence of probability measures
$(\mu_{r,t})_{r,t>0}$ on a compact space metric $X.$
Suppose additionally that:
there exists ...

**3**

votes

**1**answer

86 views

### Metric density theorem in most general setting?

It's a consequence of Lebesgue's theorem that every measurable $E\subset\mathbb{R}^n$ has a metric density that's $1$ a.e. on $E$ and $0$ a.e. on $\mathbb{R}^n\setminus E$. What are the most general ...

**1**

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**1**answer

120 views

### Ergodic decomposition and integral representation of functions that depends on a measure

Let $X$ be a compact metric space, $T:X \to X$ continuous, $M_T(X)$ the set of borel measure that are $T$-invariant and $E_T(X)\subseteq M_T(X)$ the set of ergodic measures.
The ergodic decomposition ...

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333 views

### The borel $\sigma-$algebra of the set of probability measures

Let $X$ be a compact metric space and $M(X)$ the set of all Borel probability measures on $X$.
It is know that $M(X)$ is a convex compact metric space endowed with the weak-* topology i.e.
$(\mu_n)_n ...

**8**

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**1**answer

193 views

### Besicovitch Covering Lemma on Manifolds

The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ ...

**1**

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**0**answers

81 views

### On the proof of a $W^{2,p}$ estimate - regularity on eliptic PDE

I see this proof on http://www.math.uiowa.edu/~lwang/cccalderon.pdf and I couldn't understand what he did. If $||f||_{L^p(B_{4})} = \delta$ is small and the measure $|\{ x \in B_1; ...

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175 views

### PDE-Based Triangle Inequality for Optimal Transportation

Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and ...

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**0**answers

173 views

### Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints

Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be ...

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**2**answers

185 views

### Birkhoff Ergodic Theorem or Counterexample

The Birkhoff Ergodic Theorem states:
Let $(X,\mathcal{B},m)$ be a finite or sigma finite measure space. Suppose $T:(X,\mathcal{B},m)\to (X,\mathcal{B},m)$ is measure-preserving and $f\in L^1(m)$. ...

**10**

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**1**answer

251 views

### smooth Luzin theorem

For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...

**-2**

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**1**answer

111 views

### a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...

**15**

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553 views

### Which distributions can you sample if you can sample a Gaussian?

Explicitly: You have a computer that is able to pick a real number at random according to the normal distribution: $\mathcal{N}(0,1) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Which distributions can this ...

**10**

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**1**answer

273 views

### Is this property equivalent to Lusin's property (N) for continuous functions?

A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies Lusin's (N) property if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is ...

**0**

votes

**1**answer

143 views

### Dense subsets on set space

Let $X$ be a metric space, and $\mathscr{B}$ the $\sigma$-algebra generated by open sets of $X$. Can we find a countable dense subsets of the metric space $(\mathscr{B},d)$ with the metric ...

**1**

vote

**1**answer

68 views

### When is $L^{2}(X,\mathscr{B},m)$ spearable [duplicate]

If $X$ is a metric space, $m$ is a Borel probability space on $(X,\mathscr{B})$ where $\mathscr{B}$ is the $\sigma$-algebra generated by open sets on $X$, can we prove that the space ...

**5**

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**1**answer

116 views

### Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function,
...

**5**

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**1**answer

246 views

### What it is the volume of the unit ball section of the cone of positive definite matrices?

Let $PD_{n}$ be the cone of positive definite $n \times n$ real matrices and let $B$ be the unit sphere in $n \times n$ dimensions. What is the volume of $PD_{n} \cap B$?
EDIT: Let's assume that $B$ ...

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**0**answers

135 views

### Haar measurable sets and quotient maps

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...

**0**

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**1**answer

95 views

### Is any invariant, ergodic measure with full support on an irreducible Markov shift a Markov measure?

I have this question I have been struggling with for a while. It seems rather intuitive, however, I was not able to proof it yet:
Let $\Omega = \{1,2,\cdots,N\}$ a finite alphabet, $\Sigma \subset ...

**0**

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**0**answers

70 views

### Measure generated by Semigroup $\exp[-t|p|]$

I am studying Ingrid Daubechies' paper 'An Uncertainty Principle for Fermions with Generalized
Kinetic Energy'.
On page 514, the measure $\mu_{x,y;t}$ is introduced as generated by the semigroup ...

**2**

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**0**answers

40 views

### Measurability of functions with multiple parameters

For a formalisation of the Giry monad in a theorem prover, I think I require some notion of measurability of “curried” functions. I.e. I have measure spaces $A$, $B$, and $C$ and a function $f: A ...

**2**

votes

**1**answer

85 views

### Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that
$$
\frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...

**9**

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**2**answers

305 views

### Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...

**7**

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**1**answer

209 views

### Absolute continuity reflected in Fourier coefficients?

Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$.
We say that $\mu$ is absolutely continuous with respect to $\nu$, if for every measurable set $A$ such that ...

**3**

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**0**answers

109 views

### $f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$
Let $1\leq p \leq ...

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**0**answers

306 views

### A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is motivated ...

**1**

vote

**1**answer

201 views

### question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see:
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
...

**4**

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**1**answer

159 views

### Homeomorphisms that admit a decomposition

Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$.
If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ ...

**3**

votes

**1**answer

179 views

### Multivariate monotonic function

Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ ...

**-1**

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**2**answers

131 views

### conditional expectation under convex combinaison of probability measures(II)

Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...

**1**

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**2**answers

175 views

### Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...

**4**

votes

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184 views

### Tightness of Measures, Riesz Representation for locally compact spaces

Let $X$ be a locally compact, separable metric space and $\mu_n$ a sequence of probability measures on $S$. Let $\mathfrak{C}$ be a convergence determining class for the weak topology (for instance, ...

**4**

votes

**1**answer

357 views

### Lebesgue measure of some set of irrational numbers

Let $(i_{n})$ be a strictly increasing sequence of natural numbers,
$(v_{n})$ be an unbounded sequences of natural numbers
and $M\geq 2$. Denote by $\mathcal{I}(i_{n}, v_{n}, M)$ the set of all ...

**4**

votes

**0**answers

174 views

### Do all $L^{\infty}(\mu)$ spaces have the Grothendieck property?

Consider $L_{\infty}(\Omega,\Sigma,\mu)$, where $(\Omega,\Sigma,\mu)$ is any measure space. Does it it have the Grothendieck property? If the measure space is localizable, then it is true. The ...

**3**

votes

**1**answer

143 views

### conditional expectation under convex combinaison of probability measures

Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...