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

learn more… | top users | synonyms (1)

3
votes
0answers
54 views

Homework problem about the smallest sigma algebra:

Let $\mathscr{E} \subset 2^X $, then there is a unique smallest $\sigma$-algebra containing $\mathscr{E} $. Proof(Attempt): Since $\mathscr{E} \subset 2^X $, $2^X$ is $\sigma$-algebra containing ...
0
votes
1answer
21 views

An inverse use of monotone class theorem

A family of sets $C$(or say a set of sets) is a monotone class if it is closed under countable monotone limit that is if $A_n ∈ C$, $n \geq 1$, $A_n \uparrow A$(or $A_n \downarrow A$) then $A∈C$ where ...
1
vote
1answer
26 views

What prevents the restriction of a Haar measure to a closed subgroup from being a Haar measure?

Let $\mu$ be a Haar measure on a locally compact Hausdorff topological group $G$, and let $H$ be a closed subgroup of $G$. If we restrict $\mu$ to the Borel sets of $G$ which are contained in $H$ ...
1
vote
1answer
21 views

$X$ a topological space, what are the Borel sets of a closed subset $Y$?

Let $Y$ be a closed subset of $X$. Then $\mathcal B(X)$, the set of Borel sets of $X$, is the $\sigma$-algebra generated by all closed sets of $X$. Let $E$ be the intersection of $\mathcal B(X)$ ...
1
vote
1answer
28 views

A sequence tending to zero where each element cannot be bounded by an integrable function

I am teaching myself measure theory and I working through http://homepages.uconn.edu/~rib02005/real.html. In exercise 7.3, they ask: Give an example of a sequence of non-negative functions $f_n$ ...
3
votes
1answer
37 views

If the weak derivative $\nabla u$ of $u\in L^2(\Omega)$ exists, then $\int_\Omega|\nabla u|^2=\int_\Omega|\nabla u^+|^2+\int_\Omega|\nabla u^-|^2$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded $u\in \mathcal{L}^2(\Omega)$ be weakly differentiable, i.e. $$\int_\Omega u\nabla\psi\;d\lambda^n=-\int\psi\nabla u\;d\lambda^n\;\;\;\text{for all ...
2
votes
1answer
34 views

Relation between two p-norms

While it's a well known that any two norms are equivalent for a finite dimensional normed linear space, I've been trying to derive the bounds for the case $X=\mathbb{R}^n$ and $l_p$-norms. Let $1 ...
1
vote
0answers
22 views

Boundary of surface

Let $S$ be the region in $\mathbb{R}^2$ bounded by $x$-axis, $x=1$, and $y=x$. Define $$ f(x,y) = \begin{cases} 0 &\mbox{if } x = 0 \text{ or if $x$ or $y$ is irrational} \\ 1/q & \mbox{if ...
0
votes
1answer
18 views

Regularity of limit measure and prove that $|\mu_h|\stackrel{*}{\rightharpoonup}|\mu|$

I have some questions. First of all, let $\mu_h$ a sequence of Radon measures and suppose that $\mu_h$ weakly-converge to another measure $\mu$. Now, this limit measure $\mu$ is still Borel? Is it ...
0
votes
0answers
23 views

How to prove rigorously $L[\delta(t-a)] = e^{-as}$?

In engineering mathematics (cf. Kreyszig, Ch. 6) the usaual way to prove the equality $$ L[\delta(t-a)]=e^{-as} $$ is as follows: definition : $\delta(t-a) = \lim g_m (t) $ (defined pointwise) ...
0
votes
1answer
15 views

$\mu^{*}(G)=1$ implies that $\mu^{*}(F\cap G)=\mu(F)$, $\forall F \in \mathcal{F}$…

where $\mu^{*}(G):=\inf\{\mu(F):F\in\mathcal{F}, G \subset F\}$, $(\Omega,\mathcal{F},\mu)$ a probability space. How would you prove it? Thank you.
1
vote
1answer
46 views

What is the complement of an element of a generated σ-algebra?

I'm reading Halmos' Measure Theory. As a newbie, I got confused by how to write out a complement of an element of a generated σ-algebra. For example, let $Ω$ is a set of points(such as $\mathbb N$ ...
0
votes
0answers
26 views

Condition for almost everywhere convergence to imply convergence in measure

Let $(X, \mathscr{B}, m)$ be a measure space, $f, f_n\: (n=1,2,...)$ be complex-valued measurable functions. Prove: If $m(X)<\infty,$ then $f_n\xrightarrow{n\to\infty} 0\; m-a.e. ...
2
votes
0answers
34 views

Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k ...
1
vote
1answer
26 views

Product of Borel and non-Borel set

It is true that product of Borel and non-Borel sets is non-Borel set? More precisely, I would like to know if $V $×$ \{1\}$ is Borel, where $V$ is Vitali set.
1
vote
1answer
28 views

Riesz Representation Theorem in Wikipedia vs. Rudin's RCA

In Rudin's Real & Complex Analysis theorem 2.14, the Riesz representation theorem gives (in my very rough phrasing) an injection from linear functionals on a space to positive Borel measures which ...
1
vote
0answers
25 views

Question on Borel functional calculus

I'm studying right now spectral theory of unbounded self-adjoint operators. A corollary of spectral theorem states the following: let $H$ be a (separable) Hilbert space and $(D_T, T)$ a self-adjoint ...
3
votes
2answers
189 views

A function that is bounded and measurable but not Lebesgue integrable

Could you give me concrete examples about "a function that is bounded and measurable but not Lebesgue integrable". Royden's textbook "Real analysis" says a bounded measurable function is said to be ...
0
votes
0answers
14 views

$\nabla\phi_k\stackrel{L^2}{\to}\nabla\phi\Rightarrow\langle\nabla u,\nabla\phi_k\rangle\stackrel{L^2}{\to}\langle\nabla u,\nabla\phi\rangle$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $(\phi_k)_{k\in\mathbb{N}}\subseteq L^2(\Omega)$ with ...
1
vote
1answer
21 views

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

Problem Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon>0$. Prove that $\{f_n\}$ is uniformly integrable. Background A family $\{f_n\}$ ...
0
votes
0answers
12 views

Prove that the limit of $p$-th root of the integral of the $p$-th power of a function is it's supremum [duplicate]

I'm learning about $L^p$ spaces and it seems that $|| \cdot ||_\infty$ is sort of defined in such a way that's it's essentially the supremum of a function. I'm wondering if a simpler claim can be ...
2
votes
1answer
37 views

Lebesgue integration

if $f : \mathbb{R} \to \mathbb{R}$ is continuous function which is Lebesgue integrable on $\mathbb{R}$ then show that there is sequence $x_n$ which goes to infinity and $x_n f(x_n)$ goes to $0$. ...
3
votes
1answer
86 views

Consensus division of a cake

There is a pie and two people with different tastes. The goal is to cut a piece, using two radial cuts like this: such that both people agree that the piece has a value of exactly a fraction $p$ of ...
4
votes
1answer
44 views

Adjoint of $L^{1}$ space

I have a question about $L^{p}$ spaces. Question: Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. Let us consider $f \in L^{1}(X)$ satisfying the following property: \begin{align*} \forall ...
0
votes
1answer
9 views

Describe the sigma algebra generated by $(-b,-a) \cup (a,b)$

I've been trying to solve the following question from one of my assignments. Let $\Omega=\mathbb{R}.$ Determine the $\sigma$-algebra generated by sets of the from $(-b,-a) \cup (a,b).$ My intuitive ...
2
votes
2answers
63 views

When is the image of a null set a null set?

I came upon this question here which contains the following statement: It is easy to prove that if $A \subset \mathbb{R}$ is null (has measure zero) and $f: \mathbb{R} \rightarrow \mathbb{R}$ is ...
0
votes
1answer
38 views

What happens if an uncountable collection of intervals is used in the definition of the Lebesgue outer measure?

I am reading this Wikipedia article about the Lebesgue measure. Note that the definition uses a countable number of intervals to cover the set $E\subset \mathbb R$. Do we get a different measure ...
2
votes
1answer
37 views

Complex Measures: Pushforward

Attention I added a hypothesis! Given measure spaces $\Omega$ and $\Pi$. Consider a complex measure: $$\mu:\Sigma(\Omega)\to\mathbb{C}:\quad\mu\left(\biguplus_kA_k\right)=\sum_k\mu(A_k)$$ Regard a ...
1
vote
0answers
26 views

Intersecting sets of positive Lebesgue measure with translation

If $A$ and $B$ are measurable sets of $\mathbb{R}^n$ with strictly positive but finite measure, my problem is to prove that there is a vector $c \in \mathbb{R}^n$ such that $m((A+c) \cap B) > 0$. ...
3
votes
1answer
29 views

If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous.

Background Let $E_M=\{x: |f_n(x)>M\}$. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon >0$, there exists $M$ such that $$\int_{E_M} |f_n| \ ...
1
vote
2answers
70 views

Complex Measures: Norm

Given a measure space $\Omega$. Consider a complex measure: $$\mu:\Sigma(\Omega)\to\mathbb{C}:\quad\mu\left(\biguplus_kA_k\right)=\sum_k\mu(A_k)$$ Regard the total variation: ...
0
votes
1answer
63 views

Axiomatic explanation of why the volume of a parallelepiped is equal to the area of its base times its height

Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope $P\subseteq\mathbb R^n$ a real number $V_n(P)\ge 0$ in such a ...
0
votes
0answers
15 views

Does the coarea formula hold for smooth maps with gradient bounded below?

The coarea formula for hypersurfaces in $\mathbb R^n$ can be written in two following forms: $$ \int_{\mathbb R^n} g(x) |\nabla u(y)| dx = \int_{\mathbb R} \int_{u^{-1}(t)} g(y) d\mathscr H^{n-1}(y) ...
1
vote
1answer
25 views

About equi-integrability

Suppose $\Omega\subset \mathbb R^N$ is bounded and lipschitz boundary. Suppose $u_n, u\in H^1(\Omega)$ such that $u_n\to u$ weakly in $H^1$. Then can I conclude that $$ ...
1
vote
0answers
35 views

Existence of an homeomorphism between $X$ a complete separable metric space and a subspace of $[0,1]^{\mathbb{N}}$

Result: If $X$ is a complete separable metric space then there is a $E \subset [0,1]^{\mathbb{N}}$ such that $X$ is homeomorphic to $E$ ($E$ is a $G_\delta$ set - is the intersection of denumerable ...
3
votes
1answer
56 views

Does $\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0$ imply $u\in L^2(\Omega)$?

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $u\in C^0(\Omega)$ and $(u_n)_{n\in\mathbb{N}}\subseteq C_0^0(\Omega)$ with $$\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0\tag{1}$$ Can ...
-1
votes
1answer
42 views

Show that $\frac{\cos \frac{1}{x}}{x^2}$ is not integrable on $(0,1)$

Let $\Omega=(0,1)$. How to prove that the function $$f(x)=\frac{\cos(\frac{1}{x})}{x^2}$$ is not Lebesgue integrable in $\Omega$ ?
3
votes
0answers
113 views

Using a sequence of measures to create simple functions which approximate the Radon-Nikodym derivative of the limiting measure

I have a bunch of discrete probability measures with finite support: $\mu_1,\mu_2,\dots$, which strongly converge to an absolutely continuous probability measure $\mu$ in $\mathbf{R}^2$. That is, for ...
1
vote
0answers
30 views

Caratheodory measureability of inner set function on functional closed sets

I'm struggling with the following problem: Let $X$ be a set and $\mathcal{Z}:=\{Z\subseteq X \,\big|\,\exists\,\psi\in\mathcal{C}(X)\,:\,Z=\psi^{-1}(\{0\})\}$ the family of functional closed sets ...
3
votes
2answers
95 views

Mutual information of discrete RVs which converge in distribution to a continuous RV

We have a sequence of pairs of discrete, real-valued RVs $X_n$ and $Y_n$. Each pair is characterized by a discrete probability measure on $\mathbf{R}^2$, which we will just denote $\mu_{X_n,Y_n},$ ...
1
vote
1answer
45 views

If $M$ is Borel and $M=M_1\times M_2\times\cdots \times M_n$, then $M_i$'s are Borel?

If $M$ is a nonempty Borel set in $\mathbb{R}^n$ and $M=M_1\times M_2\times\cdots \times M_n$, then are $M_1,M_2,\ldots,M_n$ are Borel sets in $\mathbb{R}$? I think the answer is yes. Using ...
3
votes
1answer
54 views

Does measurability really matter?

I am studying applied math and I currently got stuck on proving that a function, which emerges in a model is measurable (Borel functon), so we can integrate it. I know, that there are examples of ...
3
votes
2answers
39 views

Steinhaus-like problem

I know there are similar problems on here, but I believe this is not a duplicate. Let $E \subset \mathbb{R}$ be a measurable set of positive finite measure. Define $f:[0,\infty) \rightarrow ...
2
votes
1answer
27 views

Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

Problem Statement Let $\{f_n\}$ be a sequence of real-valued, measurable functions on $[0,1]$ that is uniformly bounded. Show that if $A$ is a Borel subset of $[0,1]$ then there exists subsequence ...
0
votes
2answers
36 views

Difference between topology and sigma-algebra axioms.

One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under ...
0
votes
1answer
29 views

Evaluate $ \int_{\mathbb{R}^n} \! \exp(-||x||^2) d\mu$

I have to evaluate $ \int_{\mathbb{R}^n} \! \exp(-||x||^2) d\mu$ as part of another problem. Can someone give me a hint on how to do this?
2
votes
1answer
34 views

Prove that $\sum_{k=0}^{\infty}\frac{1}{(k+p)^2}=-\int_0^1\frac{x^p}{1-x}\log x \mathrm{ d}x$

Problem Statement Prove that $$\sum_{k=1}^{\infty}\frac{1}{(k+p)^2}=-\int_0^1\frac{x^p}{1-x}\log x \ \mathrm{d}x$$ Background I just learned the limit theorems (MCT, LDCT, Fatou's Lemma). This ...
2
votes
0answers
40 views

Inner measure (inner set function) on functional closed sets

I'm struggling with the following problem: Let $X$ be a set and $\mathcal{Z}:=\{Z\subseteq X \,\big|\,\exists\,\psi\in\mathcal{C}(X)\,:\,Z=\psi^{-1}(\{0\})\}$ the family of functional closed sets ...
1
vote
1answer
32 views

The Cantor set and ternary expansions

I'm trying to prove that the Cantor set $\mathcal{C}$ contains all numbers $x \in [0,1]$ with ternary expansion $x = \sum_{k=1}^\infty \frac{a_k}{3^k}$, such that $a_k=0$ or $a_k=2$. I'm going by ...
0
votes
1answer
17 views

Prove that a sequence of measures weak-star converges to another measure

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...