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

learn more… | top users | synonyms (1)

1
vote
0answers
12 views

A problem similar to $L^2$ Fourier transform, but in the setting of Borel measure.

Problem: Let $\mu$ be a finite Borel measure on the real axis, supported on a countable set $\mathbb{Q} \subset \mathbb{R}$ (I'm not sure whether here $\mathbb{Q}$ is all rational numbers ). And let ...
1
vote
1answer
30 views

Questions on Kolmogorov Zero-One Law Proof in Williams

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book: Here are my questions: Why exactly are $\mathfrak{K}_{\infty}$ and ...
0
votes
1answer
14 views

$\lbrace \lim f_n(x) \rbrace$ is a Borel set if each $f_n$ is borel

Suppose for all $n$ that $f_n:\mathbb{R}\to \mathbb{R}$ is Borel measurable. What follows is an attempt of the proof that $\lbrace x: \lim_{n\to \infty} f_n\rbrace$ is Borel measurable, but I am a bit ...
0
votes
1answer
24 views

Book on probability theory with sigma algebra

Please suggest or recommend a book on Probability theory emphasising on sigma algebra and with Kolmogorov’s axiomatic development.
3
votes
1answer
23 views

from Carathéodory Derivative definition to the derivative of $\sin(x)$

A function $f$ is Carathéodory differentiable at $a$ if there exists a function $\phi$ which is continuous at a such that $$f(x)-f(a)=\phi(x)(x-a).$$ For $f(x) = x^n$, $\phi(x) = x^{n-1} + ...
1
vote
0answers
12 views

Product measure and splitting integral

Let $(X, A, \mu), (Y, B, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu)$, $g \in \mathcal L^1 (\nu)$. I want to show that $fg \in \mathcal L^1 (\mu \otimes \nu)$ $\int fg \ d(\mu \otimes ...
1
vote
1answer
14 views

Measure of a section is zero imply other section will be zero?

How do I see this? Consider two $\sigma$-finite measure spaces $(X,A,\mu)$ and $(Y,B,\lambda)$. And let $E \in A \otimes B$ such that $\mu(E^y)=0$ $\lambda-$a.e. Than $\lambda(E_x)=0$ $\mu$-a.e. ...
0
votes
1answer
11 views

Obtaining Essential Range and Support of a Measurable Function from Estimate

The following is an old real analysis qual problem which I cannot solve. Problem. Let $f\geq 0$ be a measurable function on $\mathbb{R}^{n}$. Suppose there exists $C>0$ such that for all ...
1
vote
2answers
51 views

Expectation of $\mathbb{E}(X^{k+1})$

I have difficulties with an old exam problem : Let $X$ be a positive random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$. Show that $$\int_0^\infty t^k ...
1
vote
0answers
12 views

Nonnegative function as limit of monotone increasing sequence (Measure Theory)

I am reading Bartle's "The Elements of Integration" and am at the part where he proves Lemma 2.11: If $f$ is a nonnegative function in $M(X,X)$, then there exists a sequence $(\phi_n )$ in $M(X,X)$ ...
1
vote
1answer
13 views

Continuous random variables and probability density function

OK, I know that a random variable $X$ from some probability space to $\mathbb R$, with some additional properties. It is discrete if it's image in $\mathbb R$ is dicrete. It is otherwise called ...
0
votes
0answers
7 views

Entropy of a 2D function versus 1D function.

I am a novice in information theory so this is more of a question seeking pointers to ideas/references to think further on the thought. I want to make concrete the idea that a function of two ...
0
votes
1answer
21 views

Set of positive measures and Banach space

In measure theory i heard recently a statement in my class, which says that the set of all (positive) measures does not make a Banach space ( whereas the set of signed measures makes up a Banach space ...
1
vote
1answer
21 views

Integration by parts formula with Lebesgue Integral and distribution function

I'm struggling to find a solution for the following problem: Let $f$ be an absolutely continuos function on [a,b], let $\mu$ be a bounded Borel measure on [a,b], and let $\Phi_\mu(t)=\mu([a,t))$ with ...
-1
votes
1answer
40 views

Why do these kind of $f(n)$'s make the limsup statement hold?

Suppose we have a function $g: \mathbb{N} \to \mathbb{N}$ s.t. $g(n) \to \infty$ as $n \to \infty$. Is it true that $g \in \{f \ | \ \limsup A_{f(n)} \subseteq \limsup A_n\}$? Suppose $g \to K$ as ...
0
votes
0answers
33 views

functions which are almost continuous.

Lusin's Theorem: Suppose $f$ is measurable and finite valued on $E$ with $E$ of finite measure. Then for every $\epsilon >0$ there exists a set $F_{\epsilon}$,with $$F_{\epsilon} \subset E, ...
0
votes
3answers
52 views

Calculate $\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}$

I have tried several methods but even I can not calculate. $$\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}\,dx$$ If anyone can help, it was part of a test and still I ...
0
votes
1answer
12 views

The image of Borel set under measurable mapping

Let $f:\mathbb R^n \to \mathbb R^n$ be a measurable mapping (assuming Lebesgue measure). What we could say about an image of Borel set $B$. Is the set $f(B)$ measurable?
1
vote
1answer
21 views

Properties of Stochastic Interval

I'm reading "Limit Thoerem for Stochastic Processes" and finding it hard to calculate the Stochastic interval.For example : In proposition 2.10,$T$ is a stopping time: If $A\in\mathcal F_0$,I need ...
1
vote
1answer
28 views

Determining if $\int f_n\to 0$ implies that $f_n\to 0$ in measure and $f_n(x)\to 0$ a.e.

If $f_n$ is a sequence of measurable functions on $(X,\mu)$ into $[0,1]$ and $\int f_n\to 0$, I am trying to prove (or disprove the following): (i) $f_n$ converges to $0$ in measure. (ii) For almost ...
1
vote
0answers
26 views

How can I show the equivalent condition for $\mu^*$-measurability

This is Exercise 4.15 from "Real Analysis for Graduate Students": Let $X$ be a set and $A$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a pre-measure on $A$ such that ...
0
votes
1answer
21 views

Problems in the integration limits to apply Fubini's theorem

If $f:(0,a)\rightarrow\mathbb{R}$ integrable function and $$g(x)=\int_{x}^a \dfrac{f(t)}{t}dt.$$ Then $g$ is integrable and $\int_{0}^a g(t)dt=\int_{0}^a f(t)dt$. I have to use Fubini's theorem but ...
0
votes
0answers
10 views

Intersection of measurable correspondences

Let $X$ be a separable metrizable space and $(S, \Sigma)$ a measurable space. Let also $\Psi _{1}: S \twoheadrightarrow X$ be a weakly measurable correspondence with nonempty compact values and $\Psi ...
0
votes
1answer
19 views

Is the product of a Schwartz function and a locally integrable function integrable?

Let $f\in S(\mathbb{R}^n)$ the space of rapidly decreasing functions on $\mathbb{R}^n$ and $g\in L_{loc}^1(\mathbb{R}^n)$. Is $fg$ integrable? Namely is it true that $$ \int |fg| <\infty. $$
1
vote
1answer
52 views

Proving a statement about probability theory

Let X be arandom variable. Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$ My answer trail: $E[|X|]=\sum_X|X|P_x(X)\lt ...
3
votes
1answer
26 views

Convergence for every measurable set

Let $(f_n)$ non-negative measurable functions such that $f_n\to f$ and $\int f_n\to \int f<\infty$. We have to prove that $\int_E f_n\to \int_Ef$ for each $E$ measurable. I know that if $f_n\to f$ ...
1
vote
1answer
41 views

some properties of $\nu$ measure

For any given function $F$ satisfying the following properties $0\le F(x)\le1,\forall x\in\mathbb R$ $F(x)\le F(y),x\le y$ $\lim_{x\to-\infty}F(x)=0,\lim_{x\to\infty}F(x)=1$ $F$ is continuous from ...
0
votes
2answers
21 views

Predual of $l^1(\Gamma)$

Let $\Gamma$ be an uncountable index set. For example $\Gamma=\mathbb R$. Let $l^1(\Gamma)$ be the set of functions with countable support and finite sum: $$ \sum_{a\in\Gamma}|f(a)|<\infty. $$ The ...
2
votes
0answers
8 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
3
votes
0answers
31 views

Why is this class closed under difference?

We have two independent random variables $X\perp Y$ involving three spaces: $(\Omega,\mathcal{A},P), (E,\mathcal{E}), (F,\mathcal{F}).$: $$X:\Omega \rightarrow E,\ Y:\Omega\rightarrow F$$ My book says ...
2
votes
1answer
28 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
1
vote
1answer
38 views

Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
0
votes
0answers
27 views

Continuous convergence [on hold]

If f_n converge pointwise to $0$ in $\mathbb{R}^d$, $\int f_n dm =1$ for every $n\in \mathbb{N}$ and $g \in L^1_m \cap C(\mathbb{R}^d,\mathbb{R})$. Then how can I prove that: \begin{equation} \int ...
1
vote
0answers
10 views

Proof of Kolmogorov zero-one law in measure-theoretic setting

I have met, in some paper, the following form of the Kolmogorov zero-one law used: If $A\subseteq 2^\Bbb N$ is a subset of Cantor space such that when $x,y\in 2^\Bbb N$ are such that $x,y$ differ ...
2
votes
1answer
24 views

Total variation distance is complete

For a given measurable space $X$, $\mathcal{P}(X)$ denotes the space of all the probability measures on $X$. The total variation distance $\rho$ on $\mathcal{P}(X)$ is defined by: for $\mu, \nu \in ...
0
votes
1answer
50 views

Question about Measure Theory [on hold]

Let $(\Omega, U, P)$ be a measure space and X be random variable and its distribution function $F_x(x)=P(\{\omega: X(\omega)\le x\})=P(-\infty , x]$ and the function F is continuous at x. If the ...
0
votes
0answers
18 views

measure equality on two sigma algebras also holds on the combined sigma algebra?

we have the following setup: $Q, P \text{ are measures on the }\sigma-\text{algebras } \mathcal{F} \text{ and } \mathcal{G} $. Let $P(A)=Q(A) \forall A\in\mathcal{F}$ and $P(B)=Q(B) \forall ...
1
vote
1answer
18 views

L^p spaces are separable and complete but not compact?

Where is the mistake in my reasoning?: Let X be a separable metric space, then for every $p\in [1,\infty)$ and for every borel measure $\mu$ on $X$: $L^p_{\mu}(X)$ is separable. Therefore by a ...
0
votes
0answers
14 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
21 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
36 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
56 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
2answers
61 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 $\left\{x_n\right\}_1^\infty$ ...
-2
votes
2answers
57 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
30 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
10 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
16 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
28 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 ...
12
votes
2answers
131 views
+50

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
21 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 ...