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

learn more… | top users | synonyms (1)

0
votes
0answers
9 views

convex function with Hessian measure $D^2 f \leqslant \lambda$ $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
1
vote
1answer
19 views

Continuity of the function defined by it's measure

The question is as follows: Let $A\subset \mathbb{R}$ be a measurable set with $0<m(A)<\infty$. Define $f$ on $\mathbb{R}$ by $f(x)=m(A\cap(-\infty,x])$. Show that this is continuous. My ...
2
votes
0answers
29 views

Convergence of the integral of a product of functions.

Let $\phi:\mathbb{R^n}\to\mathbb{R}$ be a Lebesgue-measurable function, with the property that for every $n$-dimensional cube $Q$ in $\mathbb{R^n}$, we have $$ \left|\int_{Q}\phi(x)dx ...
0
votes
1answer
37 views

A measure theory question-1 [on hold]

Let $ (\Omega, \mathcal U, P)$ be a measure space and any events $A_1, A_2, A_3 \in \mathcal{U}$ And $ B$ is defined as event of occurrence of at least one of these three events. First I need to ...
3
votes
1answer
44 views

Folland, “Real Analysis”, Chapter 5.3, Exercise 36.

Folland, "Real Analysis", Chapter 5.3, Exercise 36: Let $\mathcal{X}$ be a separable Banach space and let $\mu$ be counting measure on $\mathbf{N}$. Suppose that $\left\{x_n\right\}_1^\infty$ ...
0
votes
1answer
35 views

proving a statement based on probability theory [on hold]

Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$
-1
votes
2answers
39 views

A question related to measura space

Let a real value $X$ be a random variable and consider $\int_{\Omega}|X|dP \lt \infty $. I need to show that \begin{equation*} nP(|X|\gt n)\to_{n\to \infty} 0. \end{equation*} please help me ...
2
votes
1answer
26 views

Intuition/proof that $E(X)= \int X(w) dP = \int x d\alpha$, where $\alpha$ is the cumulative distribution function of X

Looking for more intuition/help explaining the equivalence of the following two integrals. I know that the push-forward measure, or the CDF, of a random variable $X$ on a prob. space $(\Omega, \cal ...
0
votes
0answers
9 views

Which of the convolution of integrable functions is continuous?

Let $G$ be a locally compact group with Haar measure $μ$, and let $S$ and $T⊂ G$ be two measurable subsets with $0 < μ(S) < ∞,0 < μ(T) < ∞$. Show that the map $ x → μ(S ∩ xT)$ from $G$ to ...
0
votes
0answers
12 views

Radon measure is a Haar measure

Let $G$ be a locally compact group, $H$ a dense subgroup, and $μ$ a Radon measure on $G$ such that $μ(hA) = μ(A)$ holds for every measurable set $A ⊂ G$ and every $h ∈ H$. Show that $μ$ is a (left) ...
2
votes
1answer
27 views

Let $\mathcal R$ be a $\sigma$-ring, then: $\{E\subset X; E\cap F \in \mathcal R\text{ for every } F \in \mathcal R\}$ is a $\sigma$-algebra.

I'm trying to Solve the following question: Let $X$ be a non empty set and $\mathcal R$ be a $\sigma$-ring from subset of $X$. Prove that: $$S=\{E\subset X; E\cap F \in \mathcal R\text{ for ...
4
votes
0answers
38 views

Prove that the sphere is the only closed surface in $\mathbb{R}^3$ that minimizes the surface area to volume ratio.

It is well known that a sphere minimizes the surface area to volume ratio since it reaches equality in the Isoperimetric Inequality. I'm trying to prove that no other closed surface has this property. ...
1
vote
1answer
18 views

Show intersection of two algebras are not a $\sigma$-algebra

I have the following question: $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra? I believe the answer is no. I ...
0
votes
1answer
23 views

Measure Theory (Defining Measureability)

just a very basic measure theory question from the book by Bartle. Let (a) be the statement "For every $\alpha\in\mathbb{R}$, the set $A_\alpha=\{x\in X: f(x)>\alpha\}$ belongs to X." (X is a ...
4
votes
1answer
25 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
1
vote
1answer
30 views

Intuition behind variance in terms of $L^P$ norms?

I've just started working through Varadhan's Probability lecture notes, and I was wondering if there's any intuitive connection between the variance formula and Holder's inequality/ $L^p$ norms in ...
2
votes
1answer
34 views

Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
1
vote
1answer
28 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
1
vote
2answers
34 views

Basic measure theory question about $\sigma$-algebra

Let $Y, Z$ be random variables and $G$ be a $\sigma$-algebra. Page 69 of Shreve's Stochastic Calculus for Finance II says "because both $Y$ & $Z$ are $G$-measurable, their difference $Y-Z$ is as ...
2
votes
2answers
19 views

A measure theory problem

Consider Lebesgue measure $\lambda$ restricted to the class $\mathscr B$ of Borel sets in $(0,1]$. For a fixed permutation $n_1, n_2, \dots$ of the positive integers, if $x$ has dyadic expansion ...
1
vote
1answer
15 views

If $X=\{0,1\}$, there exists an outer measure $\mu^*$ on $X$ such that $\mu^* \neq \mu^+$

Background Let $\mu^*$ be an outer measure on $X$ , $\mathcal{M}^*$ the $\sigma-$ algebra of all $\mu^*$ measurable sets, $\overline{\mu}=\mu^*\bigg|_{\mathcal{M}^*},$ and $\mu^+$ the outer measure ...
1
vote
0answers
26 views

Is the almost surely limit of measurable functions measurable in probability spaces?

Suppose we have $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_n$ a sub $\sigma$-algebra of $\mathcal{F}$. Let $(X_n)_{n=1}^\infty$ be a sequence of $\mathcal{F}_n$-measurable functions converging ...
1
vote
1answer
36 views

Let $X\subset \mathbb{R}$ be Borel measurable. Can it be that $\aleph_0 <|X|<2^{\aleph_0}$?

I want to know if every Borel measurable set in the real line has cardinality either that of the naturals or of the reals. Of course the Continuum Hypothesis is not assumed. It is clear that every ...
3
votes
1answer
33 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
0
votes
0answers
16 views

On extension of real valued functions from $(0,1)^{\infty}$ onto $[0,1]^{\infty}$

Let $\lambda$ be the standard Haar measure in $[0,1]^{\infty}$. Let $f$ be a bounded real-valued function on $(0,1)^{\infty}$ such that the set of all discontinuity points of $f$ has ...
2
votes
1answer
35 views

Borel regular measure: Approximate any measureable set by compact sets

Let $(K,\mathcal{F},\mu)$ be a measure space. Let $K$ be a compact Hausdorff space and $\mu$ be a regular finite measure. We said that it is regular if $\mu(A) = \inf\{\mu(B): B \text{ open }, ...
0
votes
0answers
26 views

Construction of a Radon measure from a certain family of compact subsets

Let $X$ be a locally compact Hausdroff space. Let $\Gamma$ be a family of compact subsets of $X$ with the following properties. 1) $\emptyset \in \Gamma$. 2) $K\cup L \in \Gamma$ whenever $K ...
1
vote
1answer
38 views

Which are the measurable sets with respect to this construction of an outer dirac-type measure?

Let $x \in \Omega$ and $\mathcal{A} \subset\mathcal{P}(X)$ an at most countable set with $\emptyset, \Omega \in \mathcal{A}$. Let $$ \delta_x: \mathcal{A} \to [0,\infty], \quad A \mapsto ...
4
votes
2answers
67 views

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null?

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<2^{\aleph_0}$, is it true that $X$ is null? Of course I am not assuming the Continuum Hypothesis. EDIT: It might be helpful to know that all ...
2
votes
2answers
20 views

Convex cone of nonnegative functions in L2 has empty interior

Convex cone $S:=\{f\in L^2(\mathbb{R},\mu):f\geq 0\}$ has empty interior in $L^2(\mathbb{R},\mu)$ when $\mu$ is Lebesgue measure. I wanted to prove it but i have major holes in my knowledge of ...
0
votes
1answer
17 views

Probability, approximation by simple functions, boundedness and non-negativity.

In my probability class various results were announced requiring that a certain random variable was bounded. The specific example that interests me is the following: if $Y$ is $\mathcal{G}$-measurable ...
1
vote
0answers
19 views

Countable unions of Vitali sets…

Let $A \subset \mathbb{R}$ be sets of positive Lebesgue measure. Let $\Gamma$ be a countable dense subgroup of the additive group $\mathbb{R}$. Consider the partition of $\mathbb{R}$ canonically ...
3
votes
1answer
30 views

Transformation shift measurable

How to prove that this transformation is measurable? $\sigma:B(n)\rightarrow{B(n)} $ $\sigma(x)(k)=x(k+1)$ $\sigma(...,x_{-1},x_{0},x_{1},...)=(...,x_{0},x_{1},x_{2},...)$ where $B(n)$ with ...
1
vote
0answers
19 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
1
vote
1answer
20 views

Fatou's Lemma conditions for strict inequality

Under what conditions do we have equality (resp. strict inequality) in Fatou's Lemma? If the sequence $f_n$ is convergent, then it is obvious that equality holds. Is it the only case? There are some ...
-2
votes
1answer
62 views

Proposition on limsup

Suppose $\exists$ function $f: \mathbb{N} \to \mathbb{N}$ s.t. as $n \to \infty$, $f(n) \to \infty$. Prove that $\forall$ events (or sets) $A_1, A_2, ..., \limsup A_{f(n)} \subseteq \limsup A_n.$ ...
0
votes
0answers
31 views

A Measure Problem on Stein's Real Analysis

I'm considering problem 5 on Stein's real analysis chapter 6 $X$ is a metric space, for any positive linear functional $l$ on $C_0 (X)$ which are the continuous functional on $X$ supported in some ...
2
votes
1answer
39 views

Haar measure on locally sigma-compact metric groups

Haar measure on locally sigma-compact metric groups $G$ is a metric group, if $G$ is a topological group meanwhile $G$ is a metric space(compatible with topology). We know that there exist a Haar ...
2
votes
0answers
39 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
1
vote
0answers
25 views

Explanation of Cramer-Wold theorem

I was trying to understand mathematically what the statement of Cramer-Wold theorem means. Intuitively, I was told that two probability distribution $P,Q \in \mathbb{R}^n$ are equivalent if all their ...
0
votes
2answers
54 views

Radon-Nikodym derivative of Measures [on hold]

Im having some trouble reconciling what I thought I learned about RN Derivatives as they relate to probability measures wikipedia,lecture notes with this blog post by John Baez mentioning it as it ...
3
votes
0answers
52 views
+100

Lebesgue Decomposition Theorem only true for Borel sets?

In Evan's book "Geometric Measure Theory and Fine Properties of Functions", we have the following two theorems: Differentiation Theorem for Radon measures. Let $\nu, \mu: \mathcal P(\Bbb R^n) \to ...
1
vote
0answers
26 views

Measurable maps in metric spaces.

i have several questions about measurability of maps with values in metric spaces : 1/ When $X$ and $Y$ are two separable metric spaces, it is easy to prove that $\mathcal{B}(X\times Y) = ...
0
votes
1answer
16 views

Can we deduce if a set is measurable, given a measurable function and a measurable space?

Let $f(x):X\rightarrow Y $, where $X$ is a measurable space. Suppose that $f$ is measurable. Let $E$ be a subset of $X$. Now, suppose that $f(E)$ is closed or clopen. Can we deduce that $E$ is a ...
1
vote
1answer
39 views

If $\Omega\subseteq\mathbb{R}^n$ is bounded, then $\int_\Omega|x-y|^{1-n}\,d\lambda < \infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded with $n\ge 2$ $\left|\;\cdot\;\right|$ be the euclidean norm $\lambda$ be the Lebesgue measure on the Borelian $\sigma$-algebra of $\mathbb{R}^n$ I ...
2
votes
1answer
22 views

Pointwise Convergence: No Diagonal Subsequence Exists?

Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a ...
0
votes
0answers
23 views

Conditionals of signed measures

My question pertains the definition of regular conditional measures of signed measures defined on product spaces. Consider a Suslin measurable space $\mathcal A=X\times Y$ with the Borel ...
0
votes
0answers
9 views

Consistent estimators/convergene in probability and slutsky

Let $m_n$ be a consistent estimator of $g(\vec\alpha)$ where $\vec\alpha = (\alpha_1,\cdots,\alpha_k)\in \mathbb{R}^k$ and $v_n$ be a consistent estimator of $f(\alpha_1,g(\vec\alpha))$. Suppose that ...
1
vote
0answers
22 views

Hölder continuity of measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function and $\mu$ the (via Stieltjes inversion formula) associated measure, which is a finite Borel measure on $\mathbb R$ and let $C(\lambda)$ be the function ($\alpha \in ...
1
vote
1answer
79 views
+50

Can the sum of two measurable functions be non-measurable if they are valued in a general normed space instead of $ \mathbb{R} $?

It's well known that the sum of measurable functions is measurable, if they are real or complex valued. However, the proofs I've seen heavily rely on the usage of the countable set of rational ...