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

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Properties of the distribution function.

Let $\Omega\subset\mathbb{R}^n$ be open. Let $f:\Omega\rightarrow\mathbb{R}$ be a measurable function and $g\in L^p(\Omega)$, for some $p\geq 1$. For a measurable function $f: ...
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Problem with infinite product measures

Given some measurable space $\left(X,\mathcal{F}\right)$ and two probability measures $\mu$ and $\nu$ on this space one can define ...
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prove that a simple function on X is measurable wrt M iff Ai are all elements of M. (full question in description)

let X be a set and M be a sigma-algebra in X. (a) prove that a simple function $s = \sum_{i=1}^{n} c_i\chi_{A_i}$ on X, where the {$c_i$} are distinct and not zero, is measurable wrt M iff $A_i$ are ...
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1answer
42 views

How does Wikipedia's definition of the Lebesgue integral relate to more common definitions?

Wikipedia presents a definition of the Lebesgue integral (of a nonnegative function $f$) that I hadn't encountered before: Let $f^*(t)=\mu \left (\{x\mid f(x)>t\} \right )$. The Lebesgue ...
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Let $X$ be any set and $M=\{\emptyset, X\}$. Prove that the class of measurable functions are exactly those functions that are constant on X.

I am attempting to solve a suggested problem while studying for my upcoming real analysis exam. Could somebody please help me with this question? Question: Let $X$ be any set and $M=\{\emptyset , ...
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If $f_{n} \in L^{\infty}$, $ \int_{0}^{1}f_{n_{k}}(x)g(x)dx \rightarrow \int_{0}^{1}f(x)g(x)dx$ for every $g \in L^1$

Supposet that $\{f_{n}\}_{n=1}^{\infty} \in L^{\infty}$. Is the following statement always true? There is a subsequence $\{n_{k}\}$ and a function $f \in L^{\infty}$ such that $$ ...
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About a countably additive vectorial measure

I need a hand with this exercise. We consider the space $([0,1],\cal{B},\mu)$, where $\mu$ is the Lebesgue measure. Let $1\leq p< \infty$. We define $F:[0,1]\to \ell^p$ as ...
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1answer
30 views

the preimage of a continuous function is measurable?

Suppose that $f(x)=|x|$ in $R^d$, then can we show that for any measurable subset $E\subset R^1$, $f^{-1}(E)$ is measurable in $R^d$? P.S. It should be noted that this is not true for general $f$ ...
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1answer
22 views

Show that the image of a zero measure set is of zero measure

I saw a topic on the subject but I did not quite understand, and it was a bit old and I didn't want to resurrect it. I am going in the right direction, I just need a little nudge. let $f: \mathbb ...
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1answer
34 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
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1answer
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Lebesgue measure unique on semiring?

in our lecture it was stated that the Lebesgue measure can be uniquely extended from a semiring to a sigma algebra by Caratheodory's theorem. Unfortunately, we did not show that it is unique on the ...
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1answer
36 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
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Is there a relationship between Haar measures and Haar wavelets, other than the name Haar?

Is there a relationship between Haar measures and Haar wavelets? The Haar wavelet does not seem to be invariant to translations, nor rotations.
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Clarification on set notation for set of points where a given sequence converges.

Prove that, given a sequence of measurable functions $\{f_{n}\}$, the set of points at which $\{f_{n}\}$ converge is measurable. My solution is to first define $f(x) = \limsup_{n \to \infty} ...
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1answer
31 views

$L^{2}$ integrability implies $L^{1}$ integrability on sets of finite measure.

Let $X$ be a measurable space with $m(X) < +\infty$. I think it's clear that if $f \in L^{2}(X)$ implies that $f \in L^{1}(X)$. But when $m(X) = +\infty$, the suppose $f(x) = \frac{1}{1+|x|}$ is ...
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Generate Borel Sigma Algebra

I want to show that the Borel Sigma-Algebra on $\mathbb{R}^n$ is generated by $ A:= \{(a_1,b_1] \times \cdots\times (a_n,b_n]; a_i,b_i \in \mathbb{R} \}$ as well as $ B:= \{(-\infty,c_1] ...
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2answers
60 views

What does this set look like? [on hold]

Consider the set of points $A \subset \mathbb{Q}^2$ in the unit square with both coordinates rational: $$A=\{(x,y) \in \mathbb{Q^2} \mid \, 0 \leq x \leq 1,\,\, 0 \leq y \leq 1 \} $$ If we color the ...
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25 views

change of variable in the integral with measure $\mu$ [on hold]

Let $f:(X,\mathbb{X}, \mu) \to Y$ a measurable function and $\nu(A)=\mu(f^{-1}(A))$. show that $$\int_Yg \,d\nu = \int_X g(f(x)) \,d\mu(x)$$
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Suppose a collection of unit vectors has measure zero on the sphere. Can ${\mathbb R}^d$ be the union of the subspaces perpendicular to the vectors?

So if a union of proper subspaces has measure zero (e.g. countably many subspacees), then ${\mathbb R}^{d}$ is not the union of these proper subspaces. But what if we have a union of $d-1$ dimensional ...
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1answer
20 views

Proof about finitely additive measure

Let $X$ be some set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Assume that $\mu:\Sigma\to[0,\infty]$ is a finitely additive measure on $\Sigma$, that is $\mu(\emptyset)=0$ $U\cap V=\emptyset ...
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1answer
43 views

Prove that volume of a ball in a polytope is very small

An exercise in a book asks to prove that for a bounded convex polytope $P\subseteq\mathbb{R}^n$ defined as an intersection of $k$ closed halfspaces and for a unit ball $B^n$ contained in $P$ the ...
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75 views

What is so special about the Lebesgue-Stieltjes measure

A measure $\lambda: B(\mathbb{R}^n) \rightarrow \overline{{\mathbb{R_{\ge 0}}}}$ that is associated with a monotone increasing and right-side continuous function $F$ is called a Lebesgue-Stieltjes ...
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Showing an operator kernel is nowhere dense…

We are doing a problem that requires us to, for fixed NONZERO element $h \in L^2[0,1]$ (as in, the function $h$ is not zero on a set of positive measure in $[0,1]$) and fixed $f \in L^2[0,1]$ with ...
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1answer
9 views

non uniform convergence of integrable functions

Let $(f_n)\subseteq L(X,\mathbb{X},\mu)$ and $f_n\longrightarrow f$, then I must show that if $\lim_{n}\int \mid f_n-f\mid=0$ then $\int\mid f\mid d\mu =\lim_n \int\mid f_n\mid d\mu$. i don't know ...
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1answer
25 views

A question on measures/densities

Let $Q,P,\nu$ be three measures on the same space such that $P\ll Q,P\ll\nu,Q\ll\nu$. Define $p=\frac{dP}{d\nu},q=\frac{dQ}{d\nu}$. Then $$\frac{1}{2}\int|p-q|d\nu=1-\int\min\{p,q\}d\nu$$ This should ...
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limitting value in p-norm

Let $f,g \in L^{p} ( \mathbb R^{n} , m)$ ; $m$ being the Lebesgue measure on $\mathbb R^{n}$ . Now , for $1\le p < \infty$ find the valoue of $lim_{|x|_{n} \to \infty} ||f(x+x_{n}) + ...
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1answer
24 views

Integration with respect to counting measure.

I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let ...
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40 views

Why we are studying Dirac measure?

If we are doing mathematics for sth worthwhile not just for fun then why do we deed to work on Dirac measure $\delta_x$ at a point $x$? For an arbitrary measure space $(X,\scr{A},\delta_x)$, where ...
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1answer
16 views

Computation of integration wrt counting measure.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ is given by $f(x)=(x^2+3)\chi_{[0,2]}(x)$. Then how do we compute $\int_{\mathbb{R}}f\,d\mu$ where $\mu$ is counting measure?
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1answer
19 views

measurable function zero everywhere. [on hold]

Let $f:\Omega\rightarrow \mathbb{R}$ be measurable and assume that $\int_Af\,d\mu=0$ whenever $A\in\mathcal{F}$.Prove that $f=0$ almost everywhere.
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1answer
20 views

Bartle - integration, monotone convergence theorem

Suppose that $(f_n) \subset M^{+}(X, \mathbb{X})$, that $(f_n)$ converges to $f$, and that $\int f d\mu=\lim \int f_n d\mu < +\infty$. Prove that $$\int_E f d\mu=\lim \int_E f_n d\mu $$ for each ...
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1answer
12 views

sequence function question [on hold]

Let $(f_n)_{n=1}^\infty$ be a sequence in $\mathcal{R}[a,b]$ such that $f_n(t)\downarrow0$ for each $t\in[a,b]$. Does it follow that $\int_a^bf\rightarrow 0$? Proof or counterexample.
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24 views

Riemann integral differentiable

Let $\mathcal{f}:[a,b]\rightarrow\mathbb{R}$ be Riemann integrable, and define $F(t)=\int_a^tf$ for $t\in[a,b]$. Recall that $F$ must be continuous but need not be differentiable. Prove that $F$ is ...
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A coherent story about (Carathéodory's version of) the construction of Lebesgue measure

I have just learned how to construct the Lebesgue measure. My text uses the Carathéodory's extension theorem which nowadays seems to be the most popular way to construct it (as I read here on MSE). ...
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Confused about the definitions of atom

I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified. Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called ...
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Is every positive function defined from $\mathbb{N}$ with real values continuous?

I wonder if every positive function defined from $\mathbb{N}$ with real values is continuous. I want to prove that a function in the measure space $(\mathbb{N},P(\mathbb{N}),\mu)$, where $\mu$ is the ...
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1answer
27 views

$f_n\rightarrow f$ in measure, and $|f_n|\le g\in L_\infty$, then $f=g$ almost everywhere

$f_n\rightarrow f$ in measure, and $|f_n|\le g\in L_\infty$, then $f=g$ almost everywhere is it true: how to prove if it is false: we need a counterexample. we have tried a lot. but got stuck. ...
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Probabilistic Turing machines as random variables

A probabilistic Turing machine (PTM) is informally described as a non-deterministic Turing machine such that ''the next movement'' is chosen with a certain probability. Suppose that the input of a PTM ...
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$f\in L_1(X) \iff \sum_{k=0}^{\infty}2^k \mu(\lbrace x\in X : f(x) \ge 2^k \rbrace)<\infty $

Let $\mu X< \infty$ and f is a nonnegative measurable function. Then $f\in L_1(X) \iff \sum_{k=0}^{\infty}2^k \mu(\lbrace x\in X : f(x) \ge 2^k \rbrace)<\infty $ To prove this for $\implies$: ...
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Hints for evaluating $\lim_{n\to\infty} \int_{[0,n]} (1-x/n)^n e^{x/3}dx$

Hints for evaluating $$\lim_{n\to\infty} \int_{[0,n]} (1-x/n)^n e^{x/3}dx$$ Clearly $\lim_{n\to\infty}(1-x/n)^n \cdot e^{x/3}=e^{-2x/3}$, but can I switch the limit and the integral?
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1answer
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$\exists E\in\mathcal{X}$ such that $\mu(E)<\infty$ and $\int_X|f|d\mu<\int_E|f|d\mu + \epsilon$

If $f$ is integrable on space X, then $\forall \epsilon >0$ $\exists E\in\mathcal{X}$ such that $\mu(E)<\infty$ and $\int_X|f|d\mu<\int_E|f|d\mu + \epsilon$ To prove this somehow I need to ...
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1answer
23 views

In the proof of that Complete convergence is equivalent to convergence a.s. under independence

Complete convergence is equivalent to convergence a.s. under independence In here, it uses the second Borel-cantelli lemma for the converse. But, it is necessary to verify that $(X_n - X)$'s are ...
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76 views

Let $D$ be a subset of $L^2[0,1]$ defined in the following way

any help with this problem, it gave me hard time: Let $D$ be a subset of $L^2[0,1]$ defined in the following way: A function $f$ belongs to $D$ if and only if f is equal almost everywhere to a ...
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1answer
30 views

Proving hint given for this Lebesgue integration question

Suppose that $\{n_{k}\}$ is an increasing sequence of positive integers and $E$ is the set of all $x \in (-\pi, \pi)$ at which $\sin(n_{k}x)$ converges. Prove that $m(E)=0$. Hint: For every subset $A$ ...
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How to show: $l^*(A)\le l^*(G_\epsilon)\le l^*(A)+\epsilon $ , $l^* $:outer measure

If $A$ s a Lebesgue measurable subset of $\mathbb{R}$ and $\epsilon\gt 0$ How to show: $\exists$ an open set $G_\epsilon \supset A$ such that $l^*(A)\le l^*(G_\epsilon)\le l^*(A)+\epsilon $, $l^* ...
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1answer
19 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
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1answer
40 views

measure $\lambda(E)=0$ or $\lambda(E)=+\infty$

Let $\mu$ be a finite measure, let $\lambda<<\mu$ ($\lambda$ is absolutely continuous wrt. $\mu$) let $P_n$,$N_n$ be a Hahn decomposition for $\lambda-n\mu$. Let $P=\cap P_n$ and $N=\cup N_n$. ...
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1answer
26 views

What is the name of this measure property?

if we have a function $f \in L^p$ sucht that $||f||_p =1$ and $m$ being a finite measure. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ ...
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1answer
17 views

$r<s$ and $L^r\subset L^s$

I want to proved the following statement: Let $0<r<s\le\infty$. Then, $L^r(\mu)\subset L^s(\mu)$ if and only if $$ \exists\ \epsilon>0:\forall E\in\mathfrak M,\text{ either } \mu(E)=0\ ...
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42 views

Problem of BIG RUDIN: Chapter 3 , Q. 5 . last part [duplicate]

Suppose $\mu (X) = 1$ & $||f||_{r} < \infty$ for some $r>0$ . Show that: $lim_{p \to 0} ||f||_{p} =$ $exp. [\int_{X} {log|f|} d\mu ]$ . Now, there are arising lot of questions: 1) How the ...