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

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2
votes
1answer
23 views

Continuous functions are locally integrable?

If $K\subset\mathbb{R}$ is compact and $f:K\rightarrow\mathbb{R}$ continuous then $f\in\mathbb{L}(K)$. In other words $f$ is integrable in $K$. So far i know that since $f$ is continuous then $f(K)$ ...
4
votes
1answer
38 views

What is a Dynkin system? ($\lambda$-system)

Until recently, all my knowledge of measure theory and Lebesgue integration are from Rudin's book, which focuses solely on the Lebesgue measure, its construction and nothing else. I have just put my ...
2
votes
0answers
60 views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
2
votes
1answer
29 views

Application of Dominated Convergence Theorem help finding a Dominating function

$$\lim_{n\to\infty}\int_0^\infty \frac{n\sin(x/n)}{x(1+x^2)}$$ I wish to use the Lebesgue Dominated Convergence theorem to solve this, but I'm having trouble finding a dominating function, $g(x)$. ...
0
votes
1answer
38 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$ a.s.? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
-3
votes
0answers
33 views

If $f = g +h$ then $\int_E f = \int_E g + \int_E h$ is independent of the choices of $g$ and $h$ [on hold]

Let $f$ be a measurable function on $E$ which can be expressed as $f = g +h$ where $g$ is a finite and integrable function over $E$ and $h$ is nonnegative over $E.$ Define $\int_{E} f = \int_E g + ...
2
votes
1answer
45 views

Proof that the Cardinality of Borel Sets on $\mathbb R$ is $c$ without using the ordinals .

I'm trying to prove that cardinality of Borel sets is $c$ without using the concept of Ordinal number ! I know that the Cardinal of Borel sets are greater than $c$ because of every point in $\mathbb ...
-1
votes
1answer
21 views

Discrete measure and piecewise function

Hi guys, can anyone please help me with why we can introduce a sectionally constant function that has support $\lambda_i, i \in \mathbb{N}$. I do not understand why we can do the part I marked with ...
0
votes
1answer
17 views

Holders Inequality: Suppose $\int_{0}^\infty x^{-2}|f|^5 dx < \infty$. Prove that $\lim_{t \to 0} t^{-\frac{6}{5}} \int_0^t f(x)dx = 0$

I discovered last night that I have an error in my proof to the following problem and I need help fixing it (or need a new solution) $$ \text{Suppose that} \int_{0}^\infty x^{-2}|f|^5 dx < \infty. ...
0
votes
1answer
33 views

Does finite expectation imply finite essential supremum?

I have a real valued function $f$ with the property that $$\mathbb{E}\big[f(X)\big] = \int f(x)\ d\mathbb{P}(x) \leq c$$ for some $c > 0$. Does this imply $$ \operatorname{ess sup } \|{f(X)}\|^2 ...
0
votes
1answer
13 views

Inequality regarding measure of function and integral of function

Let $(X,\Sigma,\mu)$ be a measure space. Let $f$ be a measurable function and $t > 0, t\in \mathbb{R}.$. Denote: $$C_f(t) = \mu \{x \in \Omega : |f(x)| \geq t \}.$$ In the first part of ...
4
votes
1answer
46 views

Question on proving tight sequences.

I was just wondering how you would go about showing that a sequence of random variables is a tight sequence. For example suppose $X_{n}$ is distributed Exponentially($\lambda_n$) how would I show that ...
0
votes
1answer
31 views

$C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$

Show that $C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$, where $\Omega$ is an open subset of $\mathbb{R^n}$. My try: Let ...
0
votes
0answers
25 views

Show that map is Borel

I was reading this paper of L. Ambrosio, S. Di Marino and G. Savare http://arxiv.org/pdf/1311.1381.pdf Under the definition of plans with barycenter in $L^q$ authors say that the map ...
0
votes
0answers
21 views

Minkowski Inequality when either $||f||_p = 0$ or $||g||_p = 0$.

I will recall that Minkowski Inequality says the following: Let $E$ be a measurable set and $p \in [1,\infty]$. If the functions $f$ and $g$ belong to $L^p(E)$, then so does their sum $f + g$ and, ...
0
votes
1answer
21 views

If $\{u_n\} \to u$ and $\{v_n\} \to v$. Show that $\{\rho(u_n, v_n)\} \to \rho(u,v)$

Let $(X,\rho)$ to be a metric space in which $\{u_n\} \to u$ and $\{v_n\} \to v$. Show that $\{\rho(u_n, v_n)\} \to \rho(u,v)$ Proof: Suppose $\{u_n\} \to u$ and $\{v_n\} \to v$. This means that ...
1
vote
0answers
33 views

Riemman-Stieltjes Integral Exercise

The truth is that I have no experience with the integral of Riemann-Stieltjes and developing a Bayesian inference problem in the book "Mathematical Statistics" by Shao, appears one of these steps, I ...
1
vote
1answer
26 views

Justifying the differentiation property of the Fourier transform

Let the Fourier transform of $f\in L^1(\Bbb R)$, denoted by $\mathcal{F}f$, be defined as $$ \mathcal{F}f(y) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ixy} f(x)\,dx.$$ An oft-quoted result ...
1
vote
1answer
23 views

Convergence of sequence of random variables 2

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
3
votes
1answer
57 views

Part of proof of the set of continuous integrable functions is dense in $L^1(\Bbb R)$

I want to prove: If $g$ belongs to $L(\Bbb R, \Bbb B, \lambda)$ and $\epsilon\gt 0$, then there exists a continuous function $f$ such that $\Vert g-f\Vert_1=\int \lvert g-f\rvert \,\text{d}\lambda \lt ...
0
votes
0answers
14 views

Locally Lipschitz $H^1$ and $L_{2}$

I have a doubt. Let $\Omega\subset \mathbb{R}^n$ and the function $f:H^1{\Omega}\to L_{2}(\Omega)$ defined by $f(u)=-|u|^{p-1}u$. I have to proof that $f$ is locally lipschitz if $2p<p^*$,with ...
2
votes
2answers
46 views

If a measurable function $f$ has zero integral over every measurable set *of finite measure*, then $f=0$ a.e.?

Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a regular measure on $X$. Suppose that $g : X \to \Bbb C$ belongs to $L^{\infty}(X)$. My question is : Is it sufficient to assume ...
0
votes
0answers
13 views

Controlling $\dot W_{k,1}$ norm of a schwarz function

If $\phi \in \mathscr{S}(\mathbb{R})$ then does it follow that there is $c$ s.t. $||\nabla^k \phi||_{L_1} \leq c^k$? From the definition I have only been able to get that this is true for all $k$ in ...
0
votes
0answers
37 views

Showing a signed Borel measure is $0$ (Practice Qual question)

Let $f: [0,1] \rightarrow \mathbb{R}$ be a continuous function. Define the signed Borel measure $\mu$ on $[0,1]$ by $d\mu = fdm$. Assume $\int_{[0,1]} x^n d\mu = 0,$ $n=0,1,2, \cdots$ Prove ...
0
votes
1answer
20 views

constructing a disjoint set

Let $F_1,F_2,\dots$ be sets in some sigma algebra, let $E_i = F_i \setminus\bigcup_{j<i} F_j$. It is true that now $E_i$ and $F_i$ are disjoint, but is it true that $\bigcup_{i=1}^n E_i = ...
0
votes
0answers
21 views

Distribution of convex combination of Bernoulli random variables

Suppose $Y_1,Y_2, \ldots $ are i.i.d Bernoulli$(p)$. What is the distribution of $$\sum_{i=1}^{\infty} \frac{Y_i}{2^i}$$ I could deal case for $p=\frac{1}{2}$ using characteristic functions but for ...
3
votes
0answers
21 views

Construction of a Borel subset $E$ of $\Bbb R$ such that both $E$ and $E^c$ has positive “density” everywhere.

Let $m$ be the Lebesgue measure on $\Bbb R$, then find $E$ Borel, such that for all $a<b, a,b\in\Bbb R$, $$m(E\cap(a,b))>0,\,m(E^c\cap(a,b))>0. $$ I couldn't find one. But after some ...
1
vote
0answers
10 views

Convergence of two Lebesgue-Stieltjes integrals

I have I have a collection of bounded variation and right-continuous functions, $(F_n)_{n \in \mathbb{N}}$, and another bounded variation and right-continuous function, $G$, which satisfy $$\sup_x ...
0
votes
0answers
13 views

Is it a change of variable?

Hi everyone: In a book I am reading, they make a sort of "substitution" like this: let $B(0,R)$ be a ball in $\mathbb{R}^{N}$ $(N\geq2)$ and $f$ a locally integrable function. Let $\mu$ be a finite ...
5
votes
0answers
63 views

Measure on $\omega$ defined in the generic extension by an atomless measure algebra is atomless

Work in Cantor space with standard probability measure $m$. Suppose we are given a sequence of measurable sets $\bar{A}=\langle A_n : n\in \omega\rangle$ and a non-principal ultrafilter $U$ and the ...
2
votes
1answer
25 views

show that continuous functions on $\mathbb{R}$ are measurable

I am trying to show this using the theorem: A function $f: \Omega \to \mathbb{R}$ is measurable if and only if $f^{-1}(E) \in \mathcal{F}$ for all borel sets $E$. The proof to show a continuous ...
2
votes
0answers
35 views

f left continuous & strictly increasing; B Borel $\implies$ f(B) Borel (or at least Lebesgue Measurable)?

How's it going? In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory ...
0
votes
1answer
75 views

Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

I am reposting a question from Math Overflow, because it seems it gets no attention. Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak ...
0
votes
0answers
14 views

In Egorov's Theorem, is almost everywhere same as point-wise?

I am studying about Egorov's Theorem. My teaching assistant said to me that Egorov's theorem is roughly like the following statement: Under two conditions which are $|E|\lt+\infty$ and ...
0
votes
1answer
30 views

A partial converse of Borel-Cantelli lemma

I'm trying to solve this problem in Prof Tao's note but no progress so far: Let $(E_n)_n$ be a sequence of events with $\inf\limits_n \mathbb P(E_n) > 0$. Show that: $\mathbb P(\sum\limits_{n \geq ...
0
votes
0answers
32 views

Interchange limit and measure

$$\lim_{n\to \infty} \mu (A_n)=\mu(\lim_{n\to\infty}A_n)$$ I came across this problem in the first course of measure theory. How is the limit of sets defined? For an analogy to limit of real numbers, ...
1
vote
1answer
20 views

Prove $\{x \in X : f(x)=g(x)+ 2 \} \in \Sigma $

Let $(X, \Sigma )$ be a measurable space and $f,g: X \rightarrow \mathbb R$ be measurable functions. Prove that $\{x \in X : f(x)=g(x)+ 2 \} \in \Sigma $. You may use the algebra of measurable ...
1
vote
1answer
34 views

General property regarding outer measure for a nested sequence of sets (measurable or not).

Let $\bigcap_{n=1}^\infty E_n=∅$ and if $\mu^*(E_n) <\infty$ and $E_{n+1} \subseteq E_n $ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) =0 $ even if each $E_n$ is a non-measurable set, where ...
3
votes
1answer
43 views

Let $f_n(x) = nx^{n-1}-(n+1)x^n$, $x\in (0, 1)$. Then $\int_{(0, 1)}\sum_{n=1}^{\infty}f_n \neq \sum_{n=1}^{\infty}\int_{(0, 1)}f_n.$

I'm learning about measure theory, specifically Lebesgue integration, and need help to understand the solution to the following problem: Let $f_n(x) = nx^{n-1}-(n+1)x^n$, $x\in (0, 1)$. Show that ...
1
vote
1answer
18 views

For $A \subseteq \mathbb R$, $\exists \Sigma$, s.t. $A \notin \Sigma$?

For any set $A \subseteq \mathbb R$, there exists a sigma algebra $\Sigma$ of subsets of $\mathbb R$ such that $A \notin \Sigma$. Is this true or false? I would think false because we can easily ...
0
votes
0answers
43 views

Prove or disprove a set function is a measure

Given the sequence of finite measures $\left\{t_n\right\}$ on a measurable space $(A, M)$ with $\sup_{n\in N}\ t_n(A)<\infty$. Let $u: M\rightarrow [0,\infty]$ be $u(E) = \sum_{n=1}^{\infty} ...
3
votes
0answers
41 views

If $\{T_n\} \to T$ and $\{u_n\} \to u$, then $\{T_n(u_n)\} \to T(u)$.

Let $X$ and $Y$ be normed linear spaces. Define $$L(X,Y) = \{T:X \to Y \ \big | \ T \text{ is bounded}\}.$$ Let $\{T_n\} \to T$ in $L(X,Y)$ and $\{u_n\} \to u$ in $X$, then $\{T_n(u_n)\} \to T(u)$ ...
1
vote
1answer
27 views

$[0,\infty]$ valued measurable function

I have a question about measure theory. Let $(X,\mathcal{M}, \mu)$ be a measure space and $f$ be a $[0,\infty]$ valued $\mathcal{M}$ measurable function on $X$. Is there $a_{k} \ge 0, A_{k} \in ...
0
votes
1answer
55 views

A Lebesgue integrable function whose absolute value is not Lebesgue integrable

Let $(\Omega, \mathcal F,\mu)$ be a measurable space. Let $f:\Omega\rightarrow \mathbb R$ be $\mathcal F$-measurable. We know that: $\int_\Omega |f|d\mu<\infty\implies\int_\Omega fd\mu<\infty ...
4
votes
1answer
36 views

Let $f: [0, 1] \to \mathbb{R}$ s.t $f(0)=f(1)=0$ then measure of $A = \{h \in [0, 1] \mid \exists x \text{ such that }f(x+h) =f(x)\} \geq 1/2$.

Let $f$ be continuous function from [0, 1] to $\mathbb{R}$ s.t $f(0)=f(1)=0$. Let $A = \{h \in [0, 1] \mid \exists x \text{ such that }f(x+h) =f(x)\}$. Show that set $A$ has Lebesgue measure $\geq ...
0
votes
0answers
18 views

Is it sensible to define the absolute value of the integral and the derivative with measures in this way?

Is it sensible to define the absolute value of the integral and the derivative with measures in this way? $\mu$ and $\nu$ are measures (functions that take a shape [a set of points] and give the ...
1
vote
1answer
41 views

Are every measure on $\mathbb{R}^{n}$ borel and/or regular?

I saw the above question in an exam paper and I am not sure how to even start. The question is true for the case of Lebesgue measure but I am not sure for arbitrary measures. I have tried looking at ...
-1
votes
1answer
47 views

how can ı solve this problem? [closed]

Suppose $f$ is in $L_1$ space of $μ$, where $μ$ is the Lebesgue measure. Prove that to each $ϵ>0$, there exists a $δ>0$ so that the Lebesgue integral of the absolute value of $f$ is less than ...
-1
votes
0answers
23 views

Show that step functions are dense in $L^1 (\Bbb R)$ [closed]

A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in $\Bbb R$. Assume $f \in L^1( \Bbb R)$, and prove that there is a sequence $\{g_n\}$ ...
1
vote
0answers
32 views

Proving $m(kE) = k\cdot m(E)$

Prove that $m(kE) = k\cdot m(E)$ given $k$ is a real number $k > 0$ $m$ is Lebesgue measure in $\mathbb{R}$ Here's my proof. Let $\{I_n\}$ be a collection of open intervals that cover $E$ $m(E) ...